hyperbolic dynamics - unistra.frirma.math.unistra.fr/~franchi/livre'.pdf · 2012-02-13 ·...

Hyperbolic Dynamicsand

Brownian Motion

an introduction

Jacques FRANCHI and Yves LE JAN

January 2012 ; preliminary version

2

CONTENTS

Introduction page 5

Summary page 6

I) The Lorentz-Mobius group PSO(1, d) page 7

I.1) Lie algebras and groups : elementary introduction page 8

I.2) The Minkowski space and pseudo-metric page 19

I.3) The Lorentz-Mobius group and its Lie algebra page 22

I.4) Two remarkable subgroups of PSO(1, d) page 25

I.5) Structure of the elements of PSO(1, d) page 28

I.6) The hyperbolic space Hd and its boundary ∂Hd page 32

I.7) Cartan and Iwasawa decompositions of PSO(1, d) page 34

I.8) Notes and comments page 35

II) Hyperbolic Geometry page 37

II.1) Hyperbolic metric page 38

II.2) Geodesics and light rays page 40

II.3) Flows and leaves page 50

II.4) Physical interpretations page 57

II.5) Poincare ball and half-space models page 61

II.6) A commutation relation page 65

II.7) Busemann function page 70

II.8) Notes and comments page 73

III) Operators and Measures page 75

III.1) Casimir operator on PSO(1, d) page 75

III.2) Laplace operator D page 78

III.3) Haar measure of PSO(1, d) page 80

III.4) The spherical Laplacian ∆Sd−1

page 90

III.5) The hyperbolic Laplacian ∆ page 93

III.6) Harmonic, Liouville and volume measures page 97

III.7) Notes and comments page 108

3

IV) Kleinian groups page 111

IV.1) Terminology page 111

IV.2) Dirichlet polyhedra page 113

IV.3) Parabolic tesselation by an ideal 2n-gon page 115

IV.4) Examples of modular groups page 121

IV.5) Notes and comments page 138

V) Measures and flows on Γ\Fd page 141

V.1) Measures of Γ-invariant sets page 142

V.2) Ergodicity page 144

V.3) A mixing theorem page 146

V.4) Poincare inequality page 151

V.5) Notes and comments page 165

VI) Basic Ito Calculus page 167

VI.1) Discrete martingales and stochastic integrals page 168

VI.2) Brownian Motion page 172

VI.3) Martingales in continuous time page 174

VI.4) The Ito integral page 179

VI.5) Ito’s Formula page 184

VI.6) The Stratonovitch integral page 195

VI.7) Notes and comments page 196

VII) Brownian motions on groups of matrices page 199

VII.1) Stochastic Differential Equations page 200

VII.2) Linear Stochastic Differential Equations page 205

VII.3) Approximation of left Brownian motions by exponentials page 220

VII.4) Lyapounov exponents page 227

VII.5) Diffusion processes page 229

VII.6) Examples of group-valued Brownian motions page 231

VII.7) Relativistic Diffusion page 250

VII.8) Notes and comments page 257

4

VIII) Central Limit Theorem for geodesics page 259

VIII.1) Dual Ad-valued left Brownian motions page 261

VIII.2) Two dual diffusions page 267

VIII.3) Spectral gap along the foliation page 269

VIII.4) Resolvent kernel and conjugate functions page 275

VIII.5) Contour deformation page 278

VIII.6) Divergence of ωf page 282

VIII.7) Sinai’s Central Limit Theorem page 288

VIII.8) Notes and comments page 298

IX) Appendix relating to geometry page 299

IX.1) Structure of pseudo-symmetric matrices page 299

IX.2) Full commutation relation in PSO(1, d) page 304

IX.3) The d’Alembertian 2 on R1,d page 308

IX.4) Core-cusps decomposition page 311

X) Appendix relating to stochastic calculus page 315

X.1) A simple construction of real Brownian motion page 315

X.2) Chaos expansion page 318

X.3) Brownian path and limiting geodesic page 321

XI) Index of notation, terms, and figures page 325

XI.1) General notation page 325

XI.2) Other notation page 325

XI.3) Index of terms page 325

XI.4) Table of figures page 325

References page 326

5

Introduction

The idea of this book is to illustrate the interplay between distinct domains ofmathematics, without assuming that the reader is a specialist of any of these domains.We view it as a minor tribute to the unity of mathematics.

Its content can be summarized under three items.

Firstly, this book provides an elementary introduction to hyperbolic geometry,based on the Lorentz group. Recall that in special relativity, space-time is equippedwith a pseudo-scalar product, which vanishes on the light cone. The Lorentz groupplays in relativistic space-time a role analogue to the rotations in Euclidean space ;the main difference is the existence of boosts, also called hyperbolic rotations. Thelight cone can be viewed as the set of straight lines which are asymptotic to the unitpseudo-sphere, namely the light rays. The hyperbolic geometry is the geometry of thispseudo-sphere. The boundary of the hyperbolic space is defined as the set of light rays.Special attention is given to the geodesic and horocyclic flows.

Secondly, this book introduces basic notions of stochastic analysis : the Wienerprocess, Ito’s stochastic integral and some elements of Ito’s calculus. This introductionallows to study linear stochastic differential equations on groups of matrices, and thendiffusion processes on homogeneous spaces. In this way are constructed in particularthe spherical and hyperbolic Brownian motions, diffusions on the stable leaves, and therelativistic diffusion.

Thirdly, quotients of the hyperbolic space under a discrete group of isometries (i.e.,in two dimensions, Riemann surfaces endowed with a negative constant curvature met-ric) are also introduced, and are the framework in which are presented some elementsof hyperbolic dynamics, especially the ergodicity of the geodesic and horocyclic flows.This book culminates with an analysis of the chaotic behaviour of the geodesic flow,which is performed using stochastic analysis methods. This main result is known asSinai’s central limit theorem. Chaotic behaviour arises from the instability of thegeodesic flow : small initial perturbations produce a large effect after some time. Thisfollows from the commutation relation between the geodesic flow and either the horo-cyclic flow or the rotations, which originates in the structure of the Lie algebra of theLorentz group. The origin of the central limit theorem is De Moivre’s theorem on theconvergence in distribution of the normalized partial sums of a coin tossing game to thebell-shaped curve. It is remarkable that such a stochastic behaviour can be observedfor the solution of a differential equation.

These methods had been presented some years ago in research articles addressed to

6

experienced readers. In this book the necessary material from group theory, geometryand stochastic analysis is presented in a self-contained and voluntarily elementary way.

Only basic knowledge of linear algebra, calculus and probability theory is required.We avoided to use or define general notions of geometry such as manifolds and bundles,except for the tangent and frame bundles of the hyperbolic space, defined via theembedding in Minkowski’s space.

Of course the reader familiar with hyperbolic geometry will traverse rapidly mostof the first five chapters. Those who know stochastic analysis will do the same withthe sixth chapter and the beginning of the seventh one.

Our approach of hyperbolic geometry is based on special relativity. The key role isplayed by the Lorentz-Mobius group PSO(1, d), the Iwasawa decomposition, commu-tation relations, the Haar measure, and the hyperbolic Laplacian. We chose to presenthyperbolic geometry via relativity to benefit from the physical intuition. Besides, thischoice allows us to present, in the stochastic analysis section, a relativistic processwhich was introduced by Dudley. We found it rather appropriate since relativity isgenerally ignored in treaties about stochastic processes.

There are a lot of good expositions of stochastic analysis. We tried to make ourexposition as short and elementary as possible, to the purpose of making it easilyavailable to analysts and geometers who could legitimately be reluctant to have to gothrough fifty pages of preliminaries before getting to the heart of the subject. Ourexposition is closer to Ito’s and McKean’s original work (see [I], [MK]).

The main results and proofs (at least in the context of this book) are printed inlarge font. The reader may first glance through the remaining part, printed in smallerfont.

Finally, some related results, which are never used to prove the main results, butcomplete the expositions of hyperbolic geometry and stochastic calculus, are given inthe appendix. For the sake of completeness, the appendix also contains a constructionof the Wiener measure.

Summary

Chapter I

The Lorentz-Mobius group PSO(1, d)

A large part of this book is centred on a careful analysis of this groupof major physical interest, to which this first chapter is mainly devoted.

It begins with an elementary and short introduction to Lie algebras,containing the basic notions which will be useful in the particular case ofPSO(1, d) : Lie groups, adjoint actions, Killing form, Lie derivatives, andtheir basic properties. All these notions are presented in the frameworkof the algebra M(d) of square matrices, and the exponential of matricesis used to define the group associated with a Lie subalgebra of M(d).Note that such group is not always a Lie group, as it is not necessarilyclosed. Nevertheless all the groups which will appear in the followingare Lie groups in the usual sense. Lie groups can be thought of astransformations of systems with finitely many degrees of freedom, e.g. acurve in SO(3) describes the motion of a solid attached to a fixed point.An infinitesimal motion is described by an element of the correspondingLie algebra.

Then the Minkowski space R1,d (which in special relativity, for d =3, describes the physical space-time) and its pseudo-metric are intro-duced, together with the Lorentz-Mobius group PSO(1, d), its Lie alge-bra so(1, d), and the space Fd of Lorentz frames, on which PSO(1, d) actsboth on the right and on the left.

7

8 CHAPTER I. THE LORENTZ-MOBIUS GROUP PSO(1, D)

We introduce then the affine subgroup Ad, generated by the first boostand the parabolic (horizontal) translations, which will play a crucial rolein the sequel, at several places. We determine the conjugacy classes ofPSO(1, d) and specify the structure of its elements.

The hyperbolic space Hd is defined as the upper sheet of the unitpseudo-sphere of the Minkowski space R1,d. Its boundary ∂Hd is madeof the light rays, i.e., the straight lines of the light cone (which areasymptotic to Hd). Physically, a point in H3 is the velocity of a materialpoint having unit mass. Poincare coordinates are defined on Hd by meansof the affine subgroup Ad.

Iwasawa’s decomposition of PSO(1, d) is based on the affine subgroupAd and the rotation subgroup SO(d), while Cartan’s decomposition ofPSO(1, d) is related to the hyperbolic metric (and to polar coordinates).

I.1 Lie algebras and groups : introduction

I.1.1 M(d) and Lie subalgebras of M(d)

We shall here consider only algebras and groups of matrices, that is,subalgebras of the basic Lie algebra M(d), the set of all d×d real squarematrices (for some integer d ≥ 2), and subgroups of the basic Lie groupGL(d), the set of all d× d real square invertible matrices, known as thegeneral linear group.

The real vector space (M(d),+, ·) is made into an algebra, called aLie algebra, by means of the Lie bracket

[M,M ′] := MM ′ −M ′M = −[M ′,M ] ,

MM ′ being the usual product of the square matrices M and M ′. The Liebracket clearly satisfies the Jacobi identity : for any matrices M,M ′,M ′′

I.1. LIE ALGEBRAS AND GROUPS : INTRODUCTION 9

we have

[[M,M ′],M ′′] + [[M ′,M ′′],M ] + [[M ′′,M ],M ′] = 0 .

The adjoint action of M(d) on itself is defined by :

ad(M)(M ′) := [M,M ′] , for any M,M ′ ∈M(d) .

The Jacobi identity can be written as follows : for any M,M ′ ∈ G ,

ad([M,M ′]) = [ad(M), ad(M ′)](

= ad(M) ad(M ′)−ad(M ′) ad(M)).

The Adjoint action of the linear group GL(d) on M(d) is by conju-gation : GL(d) 3 g 7→ Ad(g) is a morphism of groups, defined by :

Ad(g)(M) := gMg−1 , for any g ∈ GL(d) , M ∈M(d) .

A simple relation between the ad and Ad actions is as follows : for anyg ∈ GL(d),M ∈M(d), we have :

ad(Ad(g)(M)

)= Ad(g) ad(M) Ad(g)−1 . (I.1)

Indeed, for any M ′ ∈ G we have :

ad[Ad(g)(M)

](M ′) = g [M, g−1M ′g] g−1 = Ad(g) ad(M) Ad(g)−1(M ′).

A Lie algebra G of matrices is a vector subspace of M(d) which isstable under the Lie bracket [·, ·].The adjoint action of G defines then a linear map M 7→ ad(M) from G into the vector spaceof derivations on G. Indeed, the Jacobi identity is equivalent to :

ad(M)([M ′,M ′′]) = [ad(M)(M ′),M ′′] + [M ′, ad(M)(M ′′)] , for any M,M ′,M ′′ ∈ G .

The Killing form K of a Lie algebra G is the bilinear form on Gdefined (by means of the trace within G) by :

K(M ′,M ′′) := TrG(ad(M ′) ad(M ′′)) , for any M ′,M ′′ ∈ G . (I.2)


The adjoint action acts skew-symmetrically on K, meaning that :

K(ad(M)(M ′),M ′′) = −K

(M ′, ad(M)(M ′′)

), for all M,M ′,M ′′ ∈ G.

Indeed, this is equivalent to K([M,M ′],M ′′

)= K

(M, [M ′,M ′′]

)for any M,M ′,M ′′, or

using the Jacobi identity under its second formulation above, to :

TrG(ad(M ′) ad(M) ad(M ′′)

)= TrG

(ad(M) ad(M ′′) ad(M ′)

), which holds clearly.

The Killing form K on G is necessarily Ad-invariant. Indeed, for anyA,B,C ∈ G and g ∈ G, we have :

ad[Ad(g)(A)

] ad

[Ad(g)(B)

](C) = Ad(g)

(ad(A) ad(B)

[Ad(g−1)(C)

]),

that is,

ad[Ad(g)(A)

] ad

[Ad(g)(B)

]= Ad(g) ad(A) ad(B) Ad(g)−1 ,

so that

K(

Ad(g)(A),Ad(g)(B))

= Tr(

ad[Ad(g)(A)

] ad

[Ad(g)(B)

])= Tr

(ad(A) ad(B)

)= K(A,B) .

I.1.2 Basic examples of Lie algebras

1) The Lie algebra sl(d) of traceless matrices.

2) The Lie algebra so(d) of antisymmetric matrices.

3) The Lie algebra so(1, d) :=A ∈ M(d + 1)

∣∣ tA = −JAJ

, whereJ ∈ M(d + 1) denotes the diagonal matrix having diagonal entries(1,−1, . . . ,−1) (in that order). Using the Lorentz quadratic form 〈 , 〉defined on R1+d by : 〈x, x〉 := x2

0 − x21 − · · · − x2

d , we have also

so(1, d) =A ∈M(d+ 1)

∣∣ 〈Ax, x〉 = 0 for all x ∈ R1+d

.

Exercise Verify that in the above cases the Killing form is given by

K(M,M ′) = Tr(tM ×M ′).


I.1.3 The exponential map

The exponential map is defined by

exp(M) :=∑n∈N

Mn/n! , for any M ∈M(d) ,

and is C∞ from M(d) into M(d). The formula exp(M + M ′) =exp(M) exp(M ′) is correct for commuting matrices M,M ′, but does nothold in general.

Denote as usual by Tr(M) :=d∑j=1

Mjj the trace of any matrix M ∈M(d), and by det(M) its determinant. The formula

det(

exp(M))

= eTr(M) (I.3)

holds trivially for diagonalisable matrices M , and hence holds for allM ∈M(d) by density. It shows that the range of exp is included in thegeneral linear group GL(d).

We denote by 1 the unit matrix, that is, the unit element of GL(d).

Lemma I.1.3.1 The differential of Ad exp at the unit 1 is ad .Furthermore :

Ad(

exp(A))

= exp[ad(A)

]on M(d) , for any A ∈M(d) . (I.4)

Proof Consider the analytical map t 7→ Φ(t) := Ad(exp(tA)) from Rinto the space of endomorphisms onM(d). It satisfies for any real t

(dods

denoting the derivative at 0 with respect to s)

:

d

dtΦ(t) =

dods

Ad(

exp((s+ t)A

))=dods

Ad(

exp(sA) exp(tA))

=dods

Φ(s) Φ(t) = ad(A) Φ(t).


Hence, Φ solves a linear differential equation, which has the uniquesolution :

Φ(t) = exp[t ad(A)

] Φ(0) = exp

[t ad(A)

].

Proposition I.1.3.2 The differential d expM of the exponential mapat any M ∈M(d) is expressed as follows : for any B ∈M(d),

d expM(B) :=dodε

exp(M+εB) = exp(M)

(∑k∈N

1

(k + 1)!ad(−M)k(B)

).

The exponential map induces a diffeomorphism from a neighbourhood of0 in M(d) onto a neighbourhood of exp(0) = 1 in GL(d).

Nota Bene We denote the derivative at 0 with respect to ε bydodε

, here

and henceforth.

Proof We have :

d expM(B) =dodε

exp(M + εB) =∑n∈N

Yn(n+ 1)!

=∑n∈N

Zn(n+ 1)!

,

with Yn :=n∑k=0

Mn−kBMk , and Zn :=n∑k=0

Ck+1n+1 M

n−k ad(−M)k(B) .

Indeed, let us prove by induction that Yn = Zn for n ∈ N. We haveY0 = B = Z0 , and assuming Yn−1 = Zn−1 , we have :

Yn = BMn +M Yn−1 = BMn +M Zn−1

= BMn +n−1∑k=0

Ck+1n Mn−k ad(−M)k(B)


= BMn + Zn − ad(−M)n(B)−n−1∑k=0

CknM

n−k ad(−M)k(B)

= Zn +BMn −n∑k=0

CknM

n−k ad(−M)k(B)

= Zn +BMn −(

ad(−M) + [A 7→MA])n

(B) = Zn ,

since ad(−M) and the left multiplication by M commute.

Finally, the above expression with Zn entails the expression of thestatement :

d expM(B) =∑

0≤k≤n<∞

Mn−k

(n− k)!

ad(−M)k

(k + 1)!(B)

= exp(M)×(∑k≥0

ad(−M)k

(k + 1)!(B)

).

In particular, we see that d exp0 is the identity map, and then that expinduces a diffeomorphism near the origin. Exercise Using Proposition I.1.3.2 and its proof, show that the differential d expMcomputed in Proposition I.1.3.2 can be expressed alternatively as follows :

d expM(B) =∑k∈N

ad(−M)k

(k + 1)!

(exp(M)B

)

=∑k∈N

ad(M)k

(k + 1)!(B) exp(M) =

∑k∈N

ad(M)k

(k + 1)!

(B exp(M)

).

I.1.4 The group associated to a Lie subalgebra

The groups we shall consider are the subgroups of the general lineargroup GL(d) which are generated by the image (under the exponentialmap) of a Lie subalgebra of M(d).


Definition I.1.4.1 We call group associated to a Lie (sub)algebra Gthe subgroup of GL(d) generated by exp(G).

We consider again the action g 7→ Ad(g) of any such group G byconjugation, already defined in Section I.1.1.

Lemma I.1.4.2 For any g ∈ G (associated to G), Ad(g) maps G intoG.

Proof For any fixed A,B ∈ G and any real t , set :

ϕ(t) := exp(tA)B exp(−tA) ∈M(d).

This defines a C∞ map ϕ from R intoM(d), solving the order one lineardifferential equation : ϕ′ = Aϕ − ϕA , and such that ϕ(0) = B ∈ G,ϕ′(0) = [A,B] ∈ G . Hence the equation satisfied by ϕ can be solved inthe vector space G, showing by uniqueness that ϕ must be G-valued.By an immediate induction, this proves that for any A1, . . . , An, B ∈ Gwe have Ad

(exp(A1) . . . exp(An)

)(B) ∈ G , that is : Ad(g)(B) ∈ G for

any g ∈ G . The Killing form is invariant under inner automorphisms : for any

g ∈ G and M,M ′ ∈ G , we have

K(Ad(g)(M),Ad(g)(M ′)

)= K(M,M ′) . (I.5)

Indeed, by definition (I.2) of K and by Formula (I.1), we have :

K(Ad(g)(M),Ad(g)(M ′)

)= Tr

(ad[Ad(g)(M)

] ad

[Ad(g)(M ′)

])= Tr

(Ad(g) ad(M) ad(M ′) Ad(g)−1

)= Tr

(ad(M) ad(M ′)

)= K(M,M ′) .

The group G associated to a Lie subalgebra G is a Lie subgroup of GL(d) when itsatisfies the following condition : there exists a neighbourhood U of 0 in M(d) such thatexp(U) ∩G = exp(U ∩ G).


This is consistent with the usual definition of a Lie group, see for example the intro-ducing chapter of [Kn], or

([Ho], Chapter II, Section 2

). In particular, we have G = M ∈

M(d) | exp(tM) ∈ G for any real t . Moreover, any Lie subgroup G is necessarily closedin GL(d) : indeed, fixing g ∈ G , a compact neighbourhood V of 0 in M(d) on which therestriction of the exponential map is one-to-one and such that exp(V) ∩ G = exp(V ∩ G),another neighbourhood U of 0 in M(d) such that exp(−U) exp(U) ⊂ exp(V), and a se-quence (gn) ⊂ G ∩ g exp(U) which converges to g , we have g−1

0 gn ∈ exp(V) ∩ G, whenceg−1

0 gn = exp(hn), with some hn ∈ V ∩ G ; selecting a sub-sequence, we may suppose thathn → h ∈ V ∩ G, whence g = g0 exp(h) ∈ G .

On the contrary, a group G associated to a Lie subalgebra G does not need to be closedin GL(d), as the following example shows :

the one-parameter subgroup H of GL(4) defined by : H :=

exp[tA]∣∣ t ∈ R

, where

A :=

(A 00 π A

), with A :=

(0 −11 0

), does not contain −1 ∈ H, as is easily seen

by considering a sequence tn ∈ 2N + 1 approaching π modulo 2π .

Observe that, by continuity of the product and of the inverse maps, the closure in GL(d)

of a group associated to a Lie subalgebra G remains a subgroup of GL(d), which we shall call

the closed group associated to the Lie subalgebra G . This is a Lie group, since it is known(see ([Ho], Chapter VIII)

)that any closed subgroup of a Lie group is also a Lie group.

In a general and usual way, any left action of any group G on anyset S is extended to any function F on this set, by setting :

LgF (s) := F (gs) , for any (g, s) ∈ G× S .Similarly, any right action of G on S is extended to functions, bysetting :

RgF (s) := F (sg) , for any (g, s) ∈ G× S .

We shall say that a real function f on a group G associated toa Lie subalgebra G is differentiable, when its restrictions to all lines[t 7→ g exp(t A)] and [t 7→ exp(t A) g] are differentiable, for any g ∈G , A ∈ G . Then we define the right Lie derivative LAf by :

LAf(g) :=dodεf[g exp(εA)

], for any g ∈ G . (I.6)


This defines a left-invariant vector field LA on G , which means that itcommutes with left translations on G :

LA Lg = Lg LA , for all g ∈ G and A ∈ G . (I.7)

We have also

LA Rg = Rg LAd(g−1)(A) , for all g ∈ G and A ∈ G . (I.8)

Similarly, the left Lie derivative L′Af is defined by :

L′Af(g) :=dodεf[

exp(εA) g], (I.9)

and defines a right-invariant vector field L′A on G :

L′A Rg = Rg L′A , for all g ∈ G and A ∈ G ,such that

L′A Lg = Lg L′Ad(g)(A) , for all g ∈ G and A ∈ G .Furthermore we have the following formula :

LARgF (h) = L′ALhF (g) , for any A ∈ G , g, h ∈ G . (I.10)

And in the same spirit, setting f(g) := f(g−1) we have :

LAf = −L′Af . (I.11)

We say that a function f is C1 on G when all LAf exist and arecontinuous on G, or equivalently according to (I.11), when all L′Af existand are continuous on G. Similarly, f is C2 on G when all LAf are C1.

The map A 7→ LA from G onto the Lie algebra of left-invariantvector fields on G is an isomorphism of Lie algebras. We have indeedthe formula :

[LA,LB] = L[A,B] , for any A,B ∈ G . (I.12)


I.1.5 Basic examples of Lie groups

0) The Lie subgroup GL+(d) of matrices having positive determinantis the group associated to the Lie algebra M(d).

Indeed, according to Definition I.1.4.1 the group associated to the Lie algebra M(d) is the

connected component GL1(d) of 1 in GL(d), which is included in the subgroup GL+(d).

Conversely, any element A ∈ GL+(d) is arbitrarily near from a diagonalizable A′ = PDP−1 ∈GL+(d), where we can take the matrix D diagonal with positive diagonal entries, so that A′

is connected to 1 within GL+(d) by an arc [0, 1] 3 t 7→ P exp(t∆)P−1 (the matrix ∆

being diagonal and such that exp(∆) = D). This shows that A ∈ GL1(d), and then that

GL1(d) = GL+(d).

We consider the examples of Section I.1.2 again (in the same order).

1) The Lie group SL(d) of matrices having determinant one is the groupassociated to the Lie subalgebra sl(d) of traceless matrices.

Indeed, on the one hand SL(d) is clearly closed and by (I.3) we have exp[sl(d)

]⊂ SL(d).

On the other hand, the preceding example shows that any A ∈ SL(d) can be written

A = exp(M1) · · · exp(Mk) = exp(M1 − τ1

d 1)· · · exp

(Mk − τk

d 1), with τj := Tr(Mj), which

establishes the claim.

2) The Lie group SO(d) of rotation matrices is the group associated tothe Lie subalgebra so(d) of antisymmetric matrices.

Indeed, on the one hand SO(d) is obviously closed and we clearly have exp[so(d)

]⊂ SO(d).

On the other hand, using the well-known fact that any % ∈ SO(d) is conjugate in SO(d) to an

element whose non-null entries are either 1’s on the diagonal, or form diagonal planar rotation

blocks

[cosϕj − sinϕjsinϕj cosϕj

], we have directly % = exp(PA tP ) ∈ so(d), with P ∈ SO(d) and

A ∈ so(d) whose non-null entries form diagonal planar blocks

[0 −ϕjϕj 0

]. This establishes

that exp[so(d)

]= SO(d), whence the claim.

3) The Lorentz-Mobius Lie group PSO(1, d), which will be our maininterest, is the group of matrices preserving the Lorentz quadratic form


defined on R1+d by : 〈x, x〉 := x20 − x2

1 − · · · − x2d , the upper sheet of the

hyperboloid Hd :=x ∈ R1+d

∣∣ 〈x, x〉 = 1, x0 > 0

, and the orientation.This is the connected component of the unit element in the group O(1, d)of all matrices which preserve the quadratic form 〈x, x〉. In other words,this is the group made of the g ∈ SL(d + 1) such that tgJg = J and[〈x, x〉 > 0 , x0 > 0

]⇒ (gx)0 > 0 , where the diagonal matrix J is

J = diag(1,−1, . . . ,−1) ∈M(d+ 1). See Sections I.2, I.3 below.

The Lorentz-Mobius group PSO(1, d) is the group associated to the Liealgebra so(1, d), as we shall see with Remark I.5.3, which states moreprecisely that actually PSO(1, d) = exp

[so(1, d)

].

Proposition I.1.5.1 The groups PSL(2) := SL(2)/±1 and

PSO(1, 2) are isomorphic. The Lie algebras sl(2) and so(1, 2) are iso-morphic.

Proof The three matrices Y0 :=

(0 −11 0

), Y1 :=

(0 11 0

), Y2 :=

(1 00 −1

)constitute

a basis of sl(2), and we have det(x0Y0 + x1Y1 + x2Y2) = x20 − x2

1 − x22 . Since for any

g ∈ SL(2) the linear map Ad(g), acting on sl(2), preserves the determinant, and since SL(2)is connected and the morphism g 7→ Ad(g) is continuous, we see that the map Ad(g) belongsto the group O(1, 2), and even to the connected component of its unit element. Hence themorphism g 7→ Ad(g) maps the Lie group SL(2) into PSO(1, 2).

Therefore, by differentiating at the unit element and using Lemma I.1.3.1, we see that themorphism A 7→ ad(A) maps the Lie algebra sl(2) into the Lie algebra so(1, 2). Moreover, ifad(A) = 0, then A commutes with all M ∈ sl(2) and then vanishes. Hence A 7→ ad(A) isindeed an isomorphism of Lie algebras (by the Jacobi identity).

Hence Ad(SL(2)) is a neighbourhood of the unit element and a subgroup of the connectedLie group PSO(1, 2), so that it has to be the whole PSO(1, 2). Finally, if Ad(g) = 1, theng ∈ SL(2) must commute with the elements of sl(2), and then be ±1.

The elements of the group PSL(2) are identified with homographies of the Poincare half

plane R×R∗+ ≡z = x+

√−1 y ∈ C

∣∣ y > 0

, by ±(a bc d

)←→

[z 7→ az+b

cz+d

].

We could have considered matrices with complex entries as well,Md(C) instead of M(d) ≡ Md(R). In this context, considering the

I.2. THE MINKOWSKI SPACE AND PSEUDO-METRIC 19

group SL2(C) of spin-matrices, we have the following continuation ofProposition I.1.5.1.

Proposition I.1.5.2 The groups PSL2(C) := SL2(C)/±1 and

PSO(1, 3) are isomorphic. The Lie algebras sl2(C) and so(1, 3) areisomorphic.

Proof We follow the very similar proof of Proposition I.1.5.1. Let us identify the Minkowskispace R1,3 (see Section I.2 below) with the subset H2 of hermitian matrices in M2(C), by

means of the map ξ 7→(

ξ0 + ξ1 ξ2 +√−1 ξ3

ξ2 −√−1 ξ3 ξ0 − ξ1

), the determinant giving the Lorentz

quadratic form. Define the action ϕ(g) of g ∈ SL2(C) on H2 by H 7→ g H tg . Themorphism ϕ maps the group SL2(C) into PSO(1, 3).

Differentiating, we get a morphism dϕ(1) from the Lie algebra sl2(C) into the Lie algebraso(1, 3), which are both six-dimensional. The kernel of this morphism is made of thosematrices A ∈ M2(C) which have null trace and which satisfy AH + H tA = 0 for anyH ∈ H2 ; A = 0 being the only solution, the two algebras are indeed isomorphic.

Hence the range ϕ(SL2(C)) is the whole PSO(1, 3). Finally, if g ∈ SL2(C) belongs to the

kernel of ϕ , then g H tg = H for any H ∈ H2, which implies g = ±1 .

Exercise The Heisenberg group H3 is the group associated with the Lie algebra generated

by the matrices A1 :=

0 1 00 0 00 0 0

and A2 :=

0 0 00 0 10 0 0

.

Describe precisely : its Lie algebra, the computation of the exponential map, the product

formula in the group, the adjoint actions ad and Ad , and the Killing form K.

I.2 The Minkowski space and pseudo-metric

Fix an integer d ≥ 2 , and consider the Minkowski space :

R1,d :=ξ = (ξ0, . . . , ξd) ∈ R× Rd

,


endowed with the Minkowski pseudo-metric (Lorentz quadratic form) :

〈ξ, ξ〉 := ξ20 −

d∑j=1

ξ2j .

We denote by (e0, e1, . . . , ed) the canonical basis of R1,d, and we orien-tate R1,d by taking this basis as direct. We have 〈ei, ej〉 = 1i=j=0 −1i=j 6=0.

Note that e0, e1⊥ is the subspace generated by e2, . . . , ed ; weidentify it with Rd−1 ; and similarly, we identify the subspace e0⊥generated by e1, . . . , ed with Rd :

e⊥0 ≡ Rd and e0, e1⊥ ≡ Rd−1.

Note that the opposite of the pseudo-metric induces obviously the Eu-clidean metric on Rd. As usual, we shall denote by Sd−1, Sd−2 the cor-responding unit Euclidean spheres.

A vector ξ ∈ R1,d is called lightlike (or isotropic) if 〈ξ, ξ〉 = 0 , timelikeif 〈ξ, ξ〉 > 0 , positive timelike or future-directed if 〈ξ, ξ〉 > 0 and ξ0 >

0 , spacelike if 〈ξ, ξ〉 < 0 , and non-spacelike if 〈ξ, ξ〉 ≥ 0 .

The light cone of R1,d is the upper half-cone of lightlike vectors of theMinkowski space :

C :=ξ ∈ R1,d

∣∣ 〈ξ, ξ〉 = 0 , ξ0 > 0.

The solid light cone of R1,d is the convex hull of C , i.e., the upperhalf of the solid cone of timelike vectors :

C :=ξ ∈ R1,d

∣∣ 〈ξ, ξ〉 ≥ 0 , ξ0 > 0.

Lemma I.2.1 (i) No plane of R1,d is included in the solid cone ofnon-spacelike vectors.

I.2. THE MINKOWSKI SPACE AND PSEUDO-METRIC 21

(ii) For any ξ, ξ′ ∈ C , we have 〈ξ, ξ′〉 ≥√〈ξ, ξ〉〈ξ′, ξ′〉 , with equality

if and only if ξ, ξ′ are collinear.

Proof Set |ξ| :=√ξ2

0 − 〈ξ, ξ〉 , for any ξ ∈ R1,d.

(i) For non-collinear ξ, ξ′ ∈ R1,d such that 〈ξ, ξ〉 ≥ 0 and 〈ξ′, ξ′〉 ≥ 0 , we must have ξ0 6= 0(since ξ0 = 0 and 〈ξ, ξ〉 ≥ 0 imply ξ = 0, collinear to ξ′) and then⟨

ξ − (ξ′0/ξ0)ξ′, ξ − (ξ′0/ξ0)ξ′⟩

= −∣∣ξ − (ξ′0/ξ0)ξ′

∣∣2 < 0 .

(ii) For ξ, ξ′ ∈ C , we have on the one hand ξ0 ≥ |ξ| and ξ0 ξ′0 ≥ |ξ| × |ξ′| ≥

d∑j=1

ξj ξ′j ,

whence 〈ξ, ξ′〉 ≥ 0 , and on the other hand :

〈ξ, ξ′〉2 ≥(ξ0ξ′0 − |ξ||ξ′|

)2= 〈ξ, ξ〉〈ξ′, ξ′〉+

(ξ0|ξ′| − ξ′0|ξ|

)2 ≥ 〈ξ, ξ〉〈ξ′, ξ′〉 .Hence 〈ξ, ξ′〉 ≥

√〈ξ, ξ〉〈ξ′, ξ′〉 , and equality occurs if and only if (ξ1, . . . , ξd) and (ξ′1, . . . , ξ

′d)

are collinear(of same sense, by the case of equality in Schwarz inequality |ξ| |ξ′| ≥

d∑j=1

ξj ξ′j

)and ξ0|ξ′|−ξ′0|ξ| = 0 . Then |ξ| = 0 holds if and only if |ξ′| = 0 holds, in which case ξ, ξ′ are

indeed collinear. Finally, for |ξ| 6= 0 , we have (ξ′1, . . . , ξ′d) = λ(ξ1, . . . , ξd) for some positive

λ , and then ξ′0 = λ ξ0 too.

Definition I.2.2 A direct basis β = (β0, β1, . . . , βd) of R1,d satisfyingthe pseudo-orthonormality condition : 〈βi, βj〉 = 1i=j=0 − 1i=j 6=0 for0 ≤ i, j ≤ d, and such that the first component of β0 is positive, will becalled henceforth a (direct and future-directed) Lorentz frame. We setπ0(β) := β0 .

The set of all (direct and future-directed) Lorentz frames of R1,d will bedenoted henceforth by Fd.

We shall identify systematically the endomorphisms of R1,d with theirmatrices in the canonical basis (e0, e1, . . . , ed) of R1,d.


I.3 The Lorentz-Mobius group and its Lie algebra

Definition I.3.1 The Lorentz group O(1, d) is the group of endomor-phisms of R1,d which preserve the Minkowski pseudo-metric 〈·, ·〉. TheLorentz-Mobius group PSO(1, d) is the connected component of the unitmatrix in the Lorentz group O(1, d).

We shall call Lorentzian any matrix which maps a (and then any) Lorentzframe to another Lorentz frame, identified with an element of O(1, d).

The special Lorentz group SO(1, d) is the subgroup of index two inO(1, d) of endomorphisms which preserve also the orientation, so thatPSO(1, d) is the subgroup of index two in SO(1, d) of endomorphismswhich preserve the light cone C and the orientation (since the columns of

the matrix of such element g form a Lorentz frame that can be continuously drawn to the

canonical basis, as will become clear below : boosts allow to draw continuously ge0 to e0 , and

then rotations allow to draw continuously the spacelike vectors ge1 , . . . , ged to e1 , . . . , ed).

The special orthogonal group SO(d) (i.e., the rotation group of Rd)is identified with the subgroup of elements fixing the base vector e0 .Similarly, we identify the subgroup of elements fixing both base vectorse0 and e1 henceforth with SO(d− 1).

As already seen in Sections I.1.2 and I.1.5 (and as will be establishedin Remark I.5.3), we have PSO(1, d) = exp

[so(1, d)

], where

so(1, d) =A ∈M(d+ 1)

∣∣ 〈Aξ, ξ〉 = 0 for all ξ ∈ R1,d

;

so that the Lorentz-Mobius group PSO(1, d) is the group associated tothe Lie algebra so(1, d), and is a Lie group (being clearly closed). Onesays that the Lie algebra of PSO(1, d) is so(1, d).

The matrices

Ej := 〈e0, ·〉 ej − 〈ej, ·〉 e0 , for 1 ≤ j ≤ d ,

I.3. THE LORENTZ-MOBIUS GROUP AND ITS LIE ALGEBRA 23

belong to the Lie algebra so(1, d), and generate so-called boosts (orhyperbolic screws) : for any ξ ∈ R1,d, t ∈ R , 1 ≤ j ≤ d ,

etEj(ξ0, . . . , ξj, . . .) =(ξ0 ch t+ ξj sh t, . . . , ξ0 sh t+ ξj ch t, . . .

).

In special relativity, a boost corresponds to the replacement of the refe-rence frame by a frame in uniform translation, or equivalently, to anacceleration in the fixed reference frame ; see Section II.4.1 below.

The matrices

Ekl := 〈ek, ·〉 el − 〈el, ·〉 ek , for 1 ≤ k, l ≤ d ,

belong to the Lie algebra so(d) ⊂ so(1, d), and generate the subgroupSO(d). In a displayed form, we have (with for example d = 4, j = 2,k = 1, l = 3) :

Ej =

0 0 1 0 00 0 0 0 01 0 0 0 00 0 0 0 00 0 0 0 0

, Ekl =

0 0 0 0 00 0 0 1 00 0 0 0 00 −1 0 0 00 0 0 0 0

.

Proposition I.3.2 The matrices Ej , Ekl | 1 ≤ j ≤ d, 1 ≤ k < l ≤ dconstitute a pseudo-orthonormal basis of so(1, d), endowed with its Kil-ling form K. Precisely, they are pairwise orthogonal, and K(Ej, Ej) =2(d − 1) = −K(Ek`, Ek`). In particular, so(1, d) is a semisimple Liealgebra : its Killing form is non-degenerate.

Proof It is clear from its definition that so(1, d) has d(d + 1)/2 dimensions, and that theEj , Ekl above are linearly independent (this is also an obvious consequence of the computa-tion of K below), and therefore constitute a basis of so(1, d). Recall from Section I.1.1 thatthe Killing form K is defined by : K(E,E′) := Tr

(ad(E) ad(E′)

).


From the definitions above of Ej , Ekl , we have easily (for 1 ≤ i, j, k, ` ≤ d) :

ad(Ej)(Ei) = Eji , ad(Ej)(Ek`) = −ad(Ek`)(Ej) = δjkE`− δjÈk , (I.13)

ad(Ek`)(Eij) = δiÈkj − δjÈki − δikE`j + δjkEì . (I.14)

We deduce that (for 1 ≤ i, j, j′, k, ` ≤ d) :

ad(Ej′) ad(Ej)(Ei) = δjj′Ei − δij′Ej , ad(Ej′) ad(Ej)(Ek`) = δjkEj′` + δjÈkj′ ,

so thatK(Ej , Ej′) = (d− 1) δjj′ +

∑1≤k<l≤d

(δjkδj′k + δj`δj′`) = 2 (d− 1) δjj′ .

It is then clear that any ad(Ek`) ad(Ej) maps any infinitesimal boost Ei on a linear com-bination of infinitesimal rotations Ek′`′ , and any infinitesimal rotation Ek′`′ on a linear com-bination of infinitesimal boosts Ei , so that K(Ej , Ek`) = 0 for all j, k, ` .

Finally, for k < `, k′ < `′, j, i < j in 1, . . . , d, we find :

ad(Ek′`′) ad(Ek`)(Ej) = (δj`δk`′ − δjkδ``′)Ek′ − (δj`δkk′ − δjkδk′`)E`′ ,and

ad(Ek′`′) ad(Ek`)(Eij)

= (δi`δk`′ − δikδ``′)Ek′j + (δikδk′` − δi`δkk′)E`′j + (δj`δk`′ − δjkδ``′)Eik′ + (δjkδk′` − δj`δkk′)Ei`′−(δj`δi`′ − δi`δj`′)Ekk′ − (δi`δjk′ − δj`δik′)Ek`′ − (δi`′δjk − δj`′δik)Ek′` − (δik′δjk − δjk′δik)E``′ ,so that

K(Ek`, Ek′`′) = (δk′`δk`′ − δkk′δ``′)×(

2 +∑j>k′

1 +∑j>`′

1 +∑i<k′

1 +∑i<`′

1)

−(δk′`δk`′ − δk`δk′`′)(1k<k′ + 1k>k′)− (δk`δk′`′ − δkk′δ``′)(1k<`′ + 1k>`′)

−(δk`δk′`′ − δkk′δ``′)(1k′<` + 1k′>`)− (δk′`δk`′ − δk`δk′`′)(1`<`′ + 1`>`′)

= − δkk′δ``′ × (2 + 2(d− 1)− 2) = 2 (1− d) 1Ek`=Ek′`′ .

Remark I.3.3 Note that PSO(1, d) acts transitively and properly onFd. Actually, every Lorentz frame β = (β0, β1, . . . , βd) is the image ofthe canonical Lorentz frame e = (e0, e1, . . . , ed) by a unique element ofPSO(1, d), which we denote henceforth by β , so that β(ej) = βj .

Note moreover that we have : g(β) = gβ , for any g ∈ PSO(1, d) andβ ∈ Fd. This describes a left action of PSO(1, d) on Fd. We have a

right action as well, namely (β, g) 7→ βg ∈ Fd, defined by : βg := βg .

I.4. TWO REMARKABLE SUBGROUPS OF PSO(1, D) 25

I.4 Two remarkable subgroups of PSO(1, d)

SetEj := Ej + E1j , for 2 ≤ j ≤ d , (I.15)

where the matrices Ej , E1j are defined in the preceding Section I.3. As[Ej, Ek] vanishes for all 2 ≤ j, k ≤ d , the set E2 , . . . , Ed generatesa commutative Lie subalgebra of so(1, d), isomorphic to Rd−1, which weshall denote by τd−1 . By Proposition I.3.2, the Killing form K vanisheson the subalgebra τd−1 . For futur convenience, we set also E1 := E1 .

Exercise Show that as a commutative Lie subalgebra of so(1, d), τd−1 is maximal.(By letting first the Lie bracket act on e0 , verify that the linear combinations of E1 and the

Ek` which commute with Ej can use only the Ek` such that k 6= j 6= ` , and not E1.)

Definition I.4.1 (i) Set for any r ∈ R :

θr := exp[r E1] =

ch r sh r 0 0 .. 0sh r ch r 0 0 .. 00 0 1 0 .. 0.. .. .. .. .. ..

0 0 0 0 .. 1

.

(ii) For any u = (0, 0, u2, .., ud) ∈ e0, e1⊥ ≡ Rd−1, set |u|2 =

−〈u, u〉 =d∑j=2

u2j , and :

θ+u := exp

[ d∑j=2

uj Ej

]=

1 + |u|2

2 − |u|22 u2 .. .. ud|u|22 1− |u|22 u2 .. .. udu2 −u2 1 0 .. 0.. .. .. .. .. ..ud −ud 0 .. 0 1

.

θr | r ∈ R and Td−1 := θ+u |u ∈ Rd−1 are Abelian subgroups of

PSO(1, d). We shall call horizontal translations the elements of Td−1.


Note that the above displayed expression for θ+u follows at once from

the following observation :[d∑j=2

uj Ej

]2

=d∑j=2

u2j E

2j , and then

[d∑j=2

uj Ej

]3

= 0 .

Note also that θ+u+v = θ+

u θ+v , so that Td−1 is isomorphic to Rd−1.

Denote then by τd the Lie subalgebra of so(1, d) generated by τd−1

and E1 . Note that [E1, Ej] = E1,j and [E1, E1,j] = Ej , so that[E1, Ej] = Ej , and τd has dimension d .

Denote by Ad the affine subgroup of PSO(1, d) generated by thesematrices, which is also the subgroup of PSO(1, d) associated with theLie subalgebra τd of so(1, d). Note that θ+

x (e0 + e1) = (e0 + e1) for anyx ∈ Rd−1 and that θt(e0 + e1) = et (e0 + e1) for any t ∈ R. Thus allmatrices of Ad fix the particular half line R∗+(e0 + e1) of the light cone C.Note that Ad is a Lie group, since it is closed, as follows easily from thefollowing proposition I.4.3.(i) : indeed, if γ = lim

n→∞θ+xnθtn , then

γ(e0 + e1) = limn→∞

θ+xnθtn(e0 + e1) = lim

n→∞etn(e0 + e1) , so that tn → t ∈ R,

and then xn must converge to some x ∈ Rd−1, yielding γ = θ+x θt ∈ Ad.

Lemma I.4.2 For any % ∈ SO(d) and 1 ≤ j ≤ d , we have

Ad(%)(Ej) ≡ %Ej %−1 = E%(ej) := −

d∑k=1

〈%(ej), ek〉Ek ,

Proof By the commutation formula (I.13), we have : [A,Ej] = EAej =

−d∑

k=1

〈Aej, ek〉Ek for any A ∈ so(d) and 1 ≤ j ≤ d , whence by (I.4) :

Ad(exp(A))(Ej) = Eexp(A)(ej) .

TWO REMARKABLE SUBGROUPS OF PSO(1, d) 27

Proposition I.4.3 (i) Any element of the Lie subgroup Ad can bewritten θ+

x θt in a unique way, for some (t, x) ∈ R× Rd−1. Moreover,for any (t, x) ∈ R× Rd−1 we have :

θt θ+x = θ+

etx θt . (I.16)

For any z = (x, y) ∈ Rd−1 × R∗+ , set Tz ≡ Tx,y := θ+x θlog y .

We have thus Ad = Tz | z ∈ Rd−1 × R∗+, and the product formula :

Tx,y Tx′,y′ = Tx+yx′, yy′ . (I.17)

(ii) For any % ∈ SO(d− 1), we have

% θ+x %−1 = θ+

%(x) , and % θt %−1 = θt .

Proof (i) The commutation formula (I.16) follows directly from theexpressions of matrices θ+

x , θt displayed above. The existence of thedecomposition θ+

u θt follows at once. The uniqueness is clear, since θt =θ+u implies obviously t = 0, u = 0 , looking again at the expressions

displayed above for these matrices. Formula (I.17) follows at once fromFormula (I.16).

(ii) follows straightforwardly from Lemma I.4.2.

Remark I.4.4 Formula (I.17) shows that the affine subgroup Ad is isomorphic to a semi-direct product of R and Rd−1, namely the classical group of translations and dilatations ofRd−1, or equivalently, to the group of d× d triangular matrices :(

1 0x y 1d−1

)∈ GL(d)

∣∣∣∣x ∈ Rd−1(written as a column), y > 0, unit 1d−1 ∈ GL(d− 1)

,

associating

(1 0x 1d−1

)with θ+

x and

(1 00 y 1d−1

)with θlog y .


I.5 Structure of the elements of PSO(1, d)

Recall that any % ∈ SO(d) is conjugate (in SO(d)) to an element whosenon-null entries are either 1’s on the diagonal, or form diagonal planar

rotation blocks :

[cosϕj − sinϕjsinϕj cosϕj

]. This is of course well known. We

determine here the structure of elements of PSO(1, d), and classify them.

Consider the three following subgroups of PSO(1, d) : SO(d) ,

PSO(1, 1)× SO(d− 1) = ch r sh r 0

sh r ch r 00 0 R

= θrR = Rθr

∣∣∣∣ r ∈ R , R ∈ SO(d− 1)

,

and T1 × SO(d− 2) =

1 + u2

2 −u2

2 u 0u2

2 1− u2

2 u 0u −u 1 00 0 0 %

= θ+ue2% = % θ+

ue2

∣∣∣∣∣u ∈ R , % ∈ SO(d− 2)

.

Note that the eigenvalues of the element

(ch r sh rsh r ch r

)are e±r (asso-

ciated with lightlike eigenvectors), and that

1 + u2

2 −u2

2 uu2

2 1− u2

2 uu −u 1

has

the unique eigenvalue 1(associated with the unique lightlike eigenray

R+(1, 1, 0)).

Theorem I.5.1 The elements of PSO(1, d) can be classified in the fol-lowing three types (and there is no other possibility) :

STRUCTURE OF THE ELEMENTS OF PSO(1, d) 29

- Those with a timelike eigenvector are conjugated to an element ofSO(d), and will be called rotations (or elliptic elements).

- Those without any timelike eigenvector and with a unique lightlikeeigenvector are conjugated to an element of PSO(1, 1)× SO(d− 1), andwill be called boosts (or loxodromic elements).

- Those without any timelike eigenvector and with two lightlike eigenvec-tors are conjugated to an element of T1 × SO(d− 2), and will be calledparabolic elements.

In other words, using the hyperbolic space Hd and its boundary ∂Hd (defined in the followingsection I.6),

- rotations are the elements of PSO(1, d) which fix at least a point of Hd ;

- parabolic isometries are the elements of PSO(1, d) which fix no point of Hd and a uniquepoint of ∂Hd ;

- boosts are the elements of PSO(1, d) which fix no point of Hd, but two points of ∂Hd (andnot 3).

Specialising to d = 2 and using Proposition I.1.5.1, it is immediately seen that an elementof PSL(2) is a rotation, parabolic, a boost, according as the absolute value of its trace is< 2 , = 2 , > 2 , respectively.

It appears that all eigenvalues of elements of PSO(1, d) are real positive or have modulus 1.

Proof 1) Let us complexify R1,d in C1,d, endowed with the sesquilinear pseudo-norm :

〈ξ, ξ′〉 := ξ0ξ′0 −d∑j=1

ξjξ′j , and fix γ ∈ PSO(1, d), linearly extended into an isometry of C1,d,

which can be diagonalised. If γv = λv , then γv = λv , 〈v, v〉 = |λ|2〈v, v〉 , and 〈v, v〉 =λ2〈v, v〉 . Therefore we have :

either |λ| = 1 or 〈v, v〉 = 0 ,

and :either λ = ±1 or 〈v, v〉 = 0 .

As a consequence, if λ = e√−1 ϕ /∈ R is an eigenvalue, with associated eigenvector v , we

must have 〈v, v〉 = 0 . Therefore, any real eigenvector associated with λ must be lightlike.

2) Suppose there exists an eigenvalue λ such that |λ| 6= 1 , and let v denote anassociated eigenvector. Then 〈v, v〉 = 〈v, v〉 = 〈v, v〉 = 0 , which by Lemma I.2.1 forces thetwo real vectors (v + v) and (v − v)

√−1 to be linearly dependent : there exist real s, t non


vanishing both, such that s (v − v) =√−1 t (v + v), meaning that w := (s −

√−1 t) v is

real. Thus w is clearly an isotropic eigenvector associated to the eigenvalue λ , which wecan choose to belong to the light cone C . Note that this implies also that λ must be real,and even positive, since γ ∈ PSO(1, d) has to preserve C .

3) According to 2) above, consider a real eigenvalue λ = er′ 6= 1 , associated with an

eigenvector w ∈ C . Since det γ = 1 , we must have another eigenvalue λ′ having modulus6= 1 , necessarily also real and associated with some eigenvector w′ ∈ C . By Lemma I.2.1,we must have 〈w,w′〉 6= 0 , whence λ′ = 〈v, γv′〉

/〈w,w′〉 = 〈γ−1v, v′〉

/〈v, v′〉 = e−r

′. This

implies also that w,w′ are not collinear. Moreover v0 := w + 〈w,w′〉−1w′ is in Hd , andv1 := w − 〈w,w′〉−1w′ is such that 〈v1, v

′1〉 = −1 . We get then at once, for some ε = ±1 :

γv0 = (ch r)v0 + (sh r)v1 and γv1 = (sh r)v0 + (ch r)v1 .

The restriction of γ to the spacelike subspace v0, v1⊥ must be a rotation in this subspace,and then be a conjugate of some % ∈ SO(d− 1).

Furthermore, the eigenvectors different from v0±v1 are the (spacelike) eigenvectors of therotation part (as any other eigenvector could be decomposed into the sum of an eigenvectorbelonging to the timelike plane v0, v1 and of an eigenvector in v0, v1⊥ with the sameeigenvalue). In particular, γ has no timelike eigenvector.

4) Suppose there is a timelike eigenvector v0 . By 1) and 3) above, the correspondingeigenvalue λ has to be ±1 , and actually 1 , since the solid light cone is preserved by γ .

The restriction of γ to the spacelike subspace v0⊥ must be a rotation in this subspace,and then be a conjugate of some % ∈ SO(d), so that γ is a rotation.

5) We consider now the remaining possibility : eigenvalues have modulus 1, and there isno timelike eigenvector.

Consider an eigenvalue λ = e√−1 ϕ /∈ R, with associated eigenvector v . By 1) above we

must have 〈v, v〉 = 0 . Consider also the real vectors u := (v + v) and u′ := (v − v)√−1 .

We have 〈u, u〉 = 〈v, v〉+ 〈v, v〉 = 〈u′, u′〉, and

γu = (cosϕ)u+ (sinϕ)u′ and γu′ = (cosϕ)u′ − (sinϕ)u , (∗)whence 〈u, u′〉 = 〈γu, γu′〉 = cos(2ϕ) 〈u, u′〉 . Since cos(2ϕ) 6= 1 , this implies 0 = 〈u, u′〉,and then for all real s, t : 〈su + tu′, su + tu′〉 = (s2 + t2) 〈u, u〉 . Hence by Lemma I.2.1, uand u′ are either spacelike, or collinear and lightlike.

Now, if u 6= 0 and u′ = αu , using the above expressions (∗) for γu, γu′, we would haveα2 = −1 , a contradiction. Hence, u and u′ span a spacelike plane, and the restriction of γto this plane is a rotation. And the restriction of γ to u, u′⊥ is an element of PSO(1, d−2)which is as well neither a boost nor a rotation. Hence, by recursion, we are left with thecase where γ ∈ PSO(1, d− 2k) can only have ±1 as eigenvalues. Any such eigenvalue has anassociated real eigenvector, which must be non-timelike.

STRUCTURE OF THE ELEMENTS OF PSO(1, d) 31

We can then restrict γ to the orthogonal of the space spanned by spacelike eigenvectors.

We are left with the case where all eigenvectors are lightlike. Let us show now that −1 cannotbe an eigenvalue : in that case, there would be u ∈ C such that γu = −u ; choosing someu′ ∈ C non-collinear to u , we should have 〈u, u′〉 > 0 , and 〈u, γu′〉 = 〈γ−1u, u′〉 < 0 , acontradiction since γu′ ∈ C .

It follows that the eigenvector is unique, since the sum of two non-collinear eigenvectorsin C would be timelike.

Note at this stage that the multiplicity of −1 in the decomposition of γ must be even,so that the restriction of γ to the subspace spanned by spacelike eigenvectors is a rotation.

6) We are finally left with γ ∈ PSO(1, d′) (with d′ = d − 2m > 0) possessing a uniqueeigenvector v ∈ C (up to a scalar), which is associated with the eigenvalue 1. We concludethe proof by showing that d′ = 2 , and that γ is conjugate to an element of T1 .

Choose some v′ ∈ C such that 〈v, v′〉 = 12

, and set v′′ := γv′, v0 := v + v′, v1 := v − v′,u :=

√2〈v′, v′′〉 . Note that v′′ ∈ C , 〈v, v′′〉 = 1

2, 〈v1, v1〉 = −〈v0, v0〉 = −1 , 〈v0, v1〉 = 0 , and

consider v2 := (v′′ − u2v − v′)/u . We have 〈v2, v〉 = 〈v2, v′〉 = 0 , 〈v2, v2〉 = −1 , so that

we can complete (v0, v1, v2) into a Lorentz basis (v0, v1, . . . , vd′). Let γ denote the elementof PSO(1, d′) which has matrix θ+

ue2 in this basis. we have then

γv′ = v′′ = u2v + v′ + u v2 = γv′ and γv = v = γv , whence γ−1γv0 = v0 and γ−1γv1 = v1 .

Hence P := γ−1γ must belong to SO(d′− 1), i.e., have in the basis (v0, v1, . . . , vd′) a matrix

of the form

1 0 00 1 00 0 Q

, with Q ∈ SO(d′ − 1). We must also have for any λ :

(λ−1)d′+1 = det(λ1−γ) = det(λ θ+

−ue2−P ). Now, this last determinant is easily computed,by adding the first column to the second one, and then subtracting the second line to thefirst one. We get so : det(λ θ+

−ue2 −P ) = (λ−1)2 det(λ1−Q), which entails det(λ1−Q) =

(λ − 1)d′−1, whence Q = 1 and then γ = γ . Finally, since γ cannot have any spacelike

eigenvector, this forces d′ = 2 , and γ is conjugate to θ+ue2 ∈ T1 .

Remark I.5.2 Non-trivial boosts form a dense open set. The complement of this set isnegligible for the Haar measure of PSO(1, d) (defined by Formula (III.7) in Section III.3).

Proof By Theorem I.5.1, all rotations and parabolic elements have the eigenvalue 1, henceare included in the algebraic hypersurface having equation det(Tx,y %−1) = 0 (in the Iwasawacoordinates). The statement is now clear, by (III.7) and Theorem III.3.5.

Remark I.5.3 The Lorentz-Mobius group is the image of its Lie algebra under the expo-nential map : PSO(1, d) = exp

[so(1, d)

].


Proof The property for γ ∈ PSO(1, d) to belong to the range of the exponential map is

clearly stable under conjugation, and holds obviously true for SO(d),PSO(1, 1), and T1 , by

Definition I.4.1.

I.6 The hyperbolic space Hd and its boundary ∂Hd

The set of vectors having pseudo-norm 1 and positive first coordi-nate is of particular interest, and constitutes the basic model for thehyperbolic space, which appears thus most naturally in the frameworkof Minkowski’s space.

Notation Let us denote by Hd the d-dimensional hyperbolic space,defined as the positive half of the unit pseudo-sphere of R1,d, that is tosay the hypersurface of R1,d made of all vectors having pseudo-norm 1and positive first coordinate :

Hd :=ξ = (ξ0, . . . , ξd) ∈ R1,d

∣∣ 〈ξ, ξ〉 = 1, ξ0 > 0

=

(ch r) e0 + (sh r)u∣∣ r ∈ R+, u ∈ Sd−1

.

This is of course a sheet of hyperboloid. Observe that any element ofPSO(1, d) maps Hd onto Hd , as is clear by noticing that the elementsξ ∈ Hd are, among the vectors having pseudo-norm 1, those whichsatisfy 〈ξ, v〉 > 0 , for any v ∈ C .

In special relativity, an element of Hd represents the (d+ 1)-velocityof a particle of unit mass. Boosts can be interpreted as an accelerationof the particle in a fixed reference frame. See Secton II.4 below.

Proposition I.6.1 The Lie subgroup Ad acts transitively and properlyon Hd : for any p ∈ Hd , there exists a unique Tx,y ∈ Ad such thatTx,y e0 = p .

x, y are called the Poincare coordinates of p .

CARTAN AND IWASAWA DECOMPOSITIONS OF PSO(1, d) 33

In other words, we have in the canonical basis e = (e0, . . . , ed), for aunique (x, y) ∈ Rd−1 × R∗+ :

p =(y2 + |x|2 + 1

2 y

)e0 +

(y2 + |x|2 − 1

2 y

)e1 +

d∑j=2

xj

yej . (I.18)

Equivalently, the Poincare coordinates (x, y) of the point p ∈ Hd havingcoordinates (p0, . . . , pd) in the canonical basis (e0, . . . , ed) are given by :

y =1

p0 − p1=

1

〈p, e0 + e1〉, and xj =

pj

p0 − p1=−〈p, ej〉〈p, e0 + e1〉

(I.19)

for 2 ≤ j ≤ d .

The Poincare coordinates (x, y) can as well be used to parametrizethe subgroup Ad by Rd−1×R∗+ , which is usually called Poincare (upper)half-space.

Proof Formula (I.18) is merely given by the first column of the matrixTx,y (recall Definition I.4.1 and Proposition I.4.3). It is easily solved ina unique way, noticing that we must have |x|2 = y2

((p0)2 − (p1)2 − 1

),

which yields Formulas (I.19).

A light ray is a future-oriented lightlike direction, i.e., an element of :

∂Hd := C/R+ =R+ ξ

∣∣ ξ ∈ C .In the projective space of R1,d, the set of light rays ∂Hd identifies withthe boundary of Hd. Note that the Lorentz-Mobius group PSO(1, d) actson ∂Hd. Note also that for any light ray η ∈ ∂Hd, η⊥ is the hyperplanetangent to the light cone C at η (in particular, η⊥ contains η).

Note that the pseudo-metric of R1,d induces canonically a Euclideanstructure on each d-space p⊥, for any p ∈ Hd ; the inner product thereonbeing simply the opposite −〈·, ·〉 of the restriction of the pseudo-metric.


I.7 Cartan and Iwasawa decompositions of PSO(1, d)

The Cartan decomposition will be used to define the hyperbolicmetric and then (in Section III.5) polar coordinates on Hd.

Theorem I.7.1(Cartan’s decomposition of PSO(1, d)

)Any γ in

PSO(1, d) can be written : γ = % θr %′ , with r ∈ R+ and % , %′ ∈ SO(d).

Moreover, we have :

(i) r = log ‖γ‖ = log ‖γ−1‖, where ‖γ‖ := max|γ v|

/v ∈ Sd

denotes

the operator norm of γ acting on the Euclidean space Rd+1, identifiedas a vector space with R1,d ;

(ii) γ = %1 θr %′1 holds if and only if %1 = % % and %′ = % %′1 , with

% ∈ SO(d− 1) if r > 0.

Proof We must obviously have : γ e0 = % θr e0 ∈ Hd . For somer ∈ R+ and u ∈ Sd−1, we can write γ e0 = (ch r) e0 + (sh r)u. Denoteby % ∈ SO(d) the rotation acting trivially in e1, u⊥ and mapping e1

to u : we have thus %−1γ e0 = (ch r) e0 +(sh r) e1 = θr e0 , meaning that%′ := θ−1

r %−1γ ∈ SO(d).

(i) We have clearly ‖γ‖ = ‖% θr %′‖ = ‖θr‖, and, using the matrixexpression of θr , we get easily ‖θr‖ = er. Considering then the rotation%0 ∈ SO(d) fixing (e0, e3, . . . , ed) and mapping (e1, e2) to (−e1,−e2), wehave θ−r = %0 θr %0 , whence ‖γ−1‖ = ‖γ‖.

(ii) Suppose now that γ = % θr %′ = %1 θr %

′1 , and set % := %−1%1 .

We have :(ch r) e0 + (sh r) % e1 = % θr e0 = % θr %

′1 e0

= θr %′e0 = θr e0 = (ch r) e0 + (sh r) e1 ,

whence if r > 0 : % e1 = e1 , meaning that % ∈ SO(d− 1). The remainingof (ii) is obvious, since θr commutes with any % ∈ SO(d− 1).

I.8. NOTES AND COMMENTS 35

Examples The Cartan decomposition of θt ∈ A2 is given for any t < 0 by :

θt = % θ|t| % , where % e1 + e1 = % e2 + e2 = 0 and % = 1 on e1, e2⊥ .

The Cartan decomposition of θ+u ∈ A2 is given for any u ∈ R by : 1 + u2

2 −u2

2 uu2

2 1− u2

2 uu −u 1

=

1 0 00 u√

u2+4−2√u2+4

0 2√u2+4

u√u2+4

1 + u2

2u2

√u2+4 0

u2

2

√u2+4 1 + u2

2 00 0 1

1 0 0

0 −u√u2+4

2√u2+4

0 −2√u2+4

−u√u2+4

.We shall actually use mainly the following Iwasawa decomposition of

PSO(1, d), which asserts that any element of PSO(1, d) is in a uniqueway the product of an element of Ad by a rotation.

Theorem I.7.2(Iwasawa’s decomposition of PSO(1, d)

)Any γ ∈ PSO(1, d) can be written in a unique way :

γ = θ+u θt % , with θ+

u θt ∈ Ad and % ∈ SO(d) .

We shall denote by Iw and Iw the canonical projections from PSO(1, d)onto Ad and SO(d) respectively :

Iw(θ+u θt %) := θ+

u θt , Iw(θ+u θt %) := % .

Proof We must have : γ e0 = θ+u θt e0 ∈ Hd . By Proposition I.6.1,

this determines a unique θ+u θt ∈ Ad, proving the uniqueness. As to

the existence, fixing γ ∈ PSO(1, d) and using Proposition I.6.1 to getθ+u θt ∈ Ad such that θ+

u θt e0 = γ e0 , we have % := (θ+u θt)

−1γ ∈ SO(d).

I.8 Notes and comments

The reference [Kn] contains an historical account as well as a lot ofreferences on Lie Theory. See in particular [Bk], [He], [Ho], and also[DFN].


Cartan and Iwasawa decompositions are usually presented in the moregeneral context of semi-simple Lie groups, see for example [He], [KN].

The Lorentz group and the Minkowski space are fundamental in rel-ativistic mechanics (at least for d = 3), see for example [LL], [HE], [W].

There are various presentations of the hyperbolic space. See in parti-cular [Rac]. We have chosen to focus on the pseudo-sphere model, inparticular to benefit from the physical interpretation. Other models willappear in the sequel as coordinate systems (see Section II.5 below).

In general relativity the Minkowski space is replaced by manifolds lo-cally modelled on it. The bundle of frames becomes a principal PSO(1, d)bundle. The fibres of the unit tangent bundle (which describes particlespositions and (d + 1)-velocities) are hyperbolic spaces. See for example[HE], [W].

Chapter II

Hyperbolic Geometry

This chapter presents the basic notions of hyperbolic geometry, sys-tematically derived from properties of the Minkowski space and of theLorentz-Mobius group. Geodesics are given by planes intersecting thelight cone in two rays, and horospheres by affine hyperplanes parallel toa light ray and intersecting the hyperbolic space. We use only elemen-tary linear algebra within the Minkowski space R1,d to calculate intrinsicformulas for the hyperbolic distance and the distance from a point to ageodesic.

We show that any tangent vector to the hyperbolic space Hd (viewedas a subspace of Minkowski’s space) can be parametrized by its basepoint and a point at infinity (i.e., on the boundary ∂Hd). Poincarecoordinates are extended to ∂Hd. We also discuss harmonic conjugationin this framework.

Then we define the geodesic and horocyclic flows, by the right actionof the affine subgroup Ad on frames, and we give physical interpretations.We present the classical ball and upper half-space models, relate thelatter to Poincare coordinates, and prove a result which illustrates theinstability of the geodesic flow.

In this same chapter we also establish a commutation relation within

37

38 CHAPTER II. HYPERBOLIC GEOMETRY

the Lorentz-Mobius group, and specify further two particular cases,which will be crucial in the proofs of the mixing and central limit (Sinai’s)theorems (of Sections V.3 and VIII.7 respectively). Finally we introducestable leaves and the Busemann function.

II.1 Hyperbolic metric

The hyperbolic space Hd carries a natural metric, induced by R1,d,as follows. Recall that the notation β was introduced in Remark I.3.3.

Proposition II.1.1 Given two elements p, p′ of Hd, let β, β′ ∈ Fd betwo Lorentz frames such that β0 = p and β′0 = p′. Then the size r =log∥∥β−1β′

∥∥ =: dist (p, p′) of β−1β′ in its Cartan decomposition (recall

Theorem I.7.1) depends only on (p, p′). It defines a metric on Hd, calledthe hyperbolic metric. Moreover, we have dist (p, p′) = argch

[〈p, p′〉

].

Proof According to Theorem I.7.1, write β−1β′ = % θr %′. Now, if

γ, γ′ ∈ Fd are such that γ0 = p and γ′0 = p′ too, then we have γ−1β ∈SO(d) and γ′

−1β′ ∈ SO(d), so that γ−1γ′ =

(γ−1β %

)θr(%′β′

−1γ′)

de-

termines the same r ∈ R+ as β−1β′. By Theorem I.7.1(i), we haver = log

∥∥β−1β′∥∥ , and then the triangle inequality is clear. And r = 0

occurs if and only if β−1β′ ∈ SO(d), which means precisely that p = p′.Finally, we have :

argch[〈p, p′〉

]= argch

[〈β0, β

′0〉]

= argch[⟨e0, β

−1β′(e0)⟩R1,d

]= argch

[〈e0, θr e0〉

]= r .

It is obvious from Proposition II.1.1 that elements of PSO(1, d) define isometries of Hd.Actually all orientation-preserving hyperbolic isometries are obtained in this way.

II.1. HYPERBOLIC METRIC 39

Proposition II.1.2 The group of orientation-preserving hyperbolic isometries, i.e., of ori-entation-preserving isometries of the hyperbolic space Hd, is canonically isomorphic to theLorentz-Mobius group PSO(1, d).

Proof Consider first g, g′ ∈ PSO(1, d) which induce the same isometry of Hd. Theng′g−1 ∈ PSO(1, d) fixes all points of Hd, hence all points of the vector space generated byHd, meaning that this is the identity map.

Consider then an orientation-preserving isometry f of Hd. Since PSO(1, d) acts transitivelyon Hd, we can suppose that f(e0) = e0 . This implies that 〈f(ξ), e0〉 = 〈ξ, e0〉 for any ξ ∈ Hd.The projection P :=

(ξ 7→ ξ − 〈ξ, e0〉 e0

)is a bijection from Hd onto e⊥0 ≡ Rd. Consider

f := P f P−1 : Rd → Rd. For any v = P (ξ), v′ = P (ξ′) ∈ Rd, we have :⟨f(v), f(v′)

⟩=⟨

f(ξ) − 〈f(ξ), e0〉 e0, f(ξ′) − 〈f(ξ′), e0〉 e0

⟩=⟨f(ξ), f(ξ′)

⟩−⟨f(ξ), e0

⟩ ⟨f(ξ′), e0

⟩= 〈ξ, ξ′〉 −

〈ξ, e0〉〈ξ′, e0〉 = 〈v, v′〉 . Thus f ∈ SO(Rd), hence must be linear. We extend it by linearityto the whole R1,d, by setting f(e0) := e0 . We have thus an element f ∈ SO(d) ⊂ PSO(1, d),which agrees with f on Hd, since for any ξ ∈ Hd :

f(ξ) = f(〈ξ, e0〉e0 + P (ξ)

)= 〈ξ, e0〉e0 + f

(P (ξ)

)= 〈f(ξ), e0〉e0 + P

(f(ξ)

)= f(ξ) .

Finally f ∈ PSO(1, d), since by linearity the preservation of orientation and of the pseudo-

metric must extend from Hd to R1,d, and the preservation of Hd entails that of the light

cone.

Proposition II.1.3 The hyperbolic distance between q1, q2 ∈ Hd, hav-ing Poincare coordinates (x1, y1), (x2, y2) respectively, expresses as :

dist (q1, q2) = argch[〈q1, q2〉

]= argch

[|x1 − x2|2 + y2

1 + y22

2 y1 y2

]. (II.1)

Proof We apply Propositions II.1.1 and I.6.1, and observe that byFormula (I.17) we have :

〈q1, q2〉 = 〈Tx1,y1e0 , Tx2,y2

e0〉 = 〈e0 , T−1x1,y1

Tx2,y2e0〉 =

⟨e0 , Tx2−x1

y1,y2y1

e0

⟩.

The result follows at once, using Formula (I.18).


Remark II.1.4 The hyperbolic length of the line element, in the upper half-space Rd−1 ×R∗+ of Poincare coordinates, is the Euclidean one divided by the height y : ds =

√|dx|2+dy2

y.

This results indeed directly from Formula (II.1), since for small |δ|2 + ε2 we have :

dist(Tx,y e0 , Tx+δ,y+ε e0

)= argch

[1 +

|δ|2 + ε2

2 y (y + ε)

]∼√|δ|2 + ε2

y (y + ε)∼√|δ|2 + ε2

y.

II.2 Geodesics and light rays

II.2.1 Hyperbolic geodesics

Definition II.2.1.1 Let us call geodesic of the hyperbolic space Hd anynon-empty intersection of Hd with a vector plane of R1,d. Thus, the setof geodesics of Hd identifies with the set of vector planes of R1,d whichintersect Hd.

The following remark justifies the above definition, and yields a naturalidentification between the set of geodesics of Hd and the set of pairsη, η′ of distinct η, η′ ∈ ∂Hd, or between the set of oriented geodesicsof Hd and the set of ordered pairs (η, η′) of distinct η, η′ ∈ ∂Hd.

Indeed, for η 6= η′ ∈ ∂Hd, and for non-null ξ ∈ η, ξ ∈ η′, we have 〈ξ, ξ′〉 > 0 by Lemma I.2.1,

and then (2〈ξ, ξ′〉)−1(ξ + ξ′) ∈ Hd : the plane cone generated by η, η′ does intersect Hd.

Remark II.2.1.2 There exists a unique geodesic containing any twogiven distinct points of Hd ∪ ∂Hd.

The geodesic containing the two points of ∂Hd fixed by a boost is called its axis.

Proposition II.2.1.3 In two dimensions, consider two oriented distinct geodesics γ, γ′ ofH2, determined (up to orientation) by distinct planes P, P ′ respectively.

- If P ∩ P ′ is timelike, then there exists a unique isometry ψ ∈ PSO(1, 2) mapping γ to γ′

and such that ψ(P ∩ P ′) = P ∩ P ′, and ψ is a rotation.

II.2. GEODESICS AND LIGHT RAYS 41

- If P ∩ P ′ is spacelike, then there exists a unique isometry ψ ∈ PSO(1, 2) mapping γ to γ′

and such that ψ(P ∩P ′) = P ∩P ′, and ψ is a boost. Moreover, there exists a unique geodesicγ′′ intersecting γ and γ′, and perpendicular to both of them, and γ′′ is the axis of the boostψ .

- If P ∩ P ′ is lightlike, then there exists a unique parabolic isometry ψ ∈ PSO(1, 2) mappingthe non-oriented γ to the non-oriented γ′ and such that ψ(P ∩ P ′) = P ∩ P ′.

There is no longer uniqueness in larger dimensions.

Proof If P ∩ P ′ is spacelike, its orthogonal is a plane intersecting H2 (it must containa timelike vector), hence a geodesic, clearly perpendicular to both P and P ′. Consider aLorentz frame β such that β2 ∈ P ∩ P ′ and β0 ∈ P . P ′ contains p ∈ H2 ∩ (P ∩ P ′)⊥,necessarily of the form : p = ch r β0 + sh r β1 with r ∈ R∗. Any solution ψ must map β0

to p, and β2 to ±β2, whence a unique possibility, depending on the orientations of γ, γ′.Obviously, the boost having matrix θr in the frame β is the unique solution, and its axis is(P ∩ P ′)⊥, as claimed.

If P ∩ P ′ is timelike, any solution is a rotation according to Theorem I.5.1, which in anyLorentz frame β such that β0 ∈ P ∩ P ′ must have its matrix in SO(2). Obviously, a uniqueone meets the case, taking the orientations of γ, γ′ into account.

If P ∩P ′ is lightlike, consider a Lorentz frame β such that β0 ∈ P and β0 +β1 ∈ P ∩P ′.We can write the ends of γ, γ′ respectively

R+(β0 + β1),R+(β0 − β1)

and

R+(β0 + β1),R+(β0 − cosϕβ1 + sinϕβ2)

, with sinϕ 6= 0 . Then the parabolic isometry ψ

having matrix θ+tg (ϕ/2) in the frame β meets the case. If ψ′ is another solution, then ψ−1ψ′

fixes R+(β0 +β1) and R+(β0−β1), and then (by Theorem I.5.1) is a boost, which must have

matrix θr in the frame β. Thus ψ′ has in the frame β matrix θ+tg (ϕ/2)θr , which happens to

have eigenvalues 1, er, (1 + tg 2(ϕ/2))e−r. By Theorem I.5.1, since ψ′ must be parabolic, this

forces r = 0 , hence ψ′ = ψ .

II.2.2 Projection onto a light ray, and tangent bundle

We have the following useful projection, from the hyperbolic spaceonto a light ray. See Figure II.1 (note that the location of p in thisfigure is generic, any analogous figure with another location of p beingmerely deduced by an isometry).


d

Figure II.1: projection from Hd onto a light ray η

Proposition II.2.2.1 For any (p, η) ∈ Hd×∂Hd, there exists a unique(pη, ηp) ∈ R1,d × η such that

〈p, pη〉 = 0 , 〈pη, pη〉 = −1 , and p+ pη = ηp (6= 0).

Proof Consider p′ := p − α η0 , for any given η0 ∈ η ∩ C . Then〈p, η0〉 > 0 , and 〈p, p′〉 = 0 ⇔ 〈p′, p′〉 = −1 ⇔ α 〈p, η0〉 = 1 showsthat there is indeed a unique solution.

Reciprocally, if q belongs to the unit sphere of p⊥, i.e., if 〈p, q〉 = 0and 〈q, q〉 = −1 , then η := R+(p+ q) ∈ ∂Hd, and pη = q .

Notation For any p ∈ Hd, denote by Jp the one-to-one map from∂Hd into p⊥ defined by : Jp(η) = pη . Its range is the unit sphere of theEuclidean d-space p⊥.

Let us denote by T 1Hd the unit tangent bundle of Hd, defined by :

T 1Hd :=

(p, q) ∈ Hd × R1,d∣∣∣ q ∈ p⊥, 〈q, q〉 = −1

.


Accordingly, Fd is known as the frame bundle of Hd.

Note that p⊥ is the vector hyperplane directing the affine hyperplanetangent to Hd at p .

Let us identify Hd × ∂Hd with the unit tangent bundle T 1Hd, bymeans of the bijection J , defined by : J(p, η) := (p, pη) =

(p, Jp(η)

).

The projection π0 (recall Definition I.2.2) goes from Fd onto Hd. Wealso define a projection π1 from Fd onto Hd×∂Hd, by : for any β ∈ Fd,

π1(β) :=(β0, R+(β0+β1)

)=(π0(β), J−1

β0(β1)

)= J−1(β0, β1) ∈ Hd×∂Hd.

Note that by the above definitions, using the right action of PSO(1, d)on Fd (recall Remark I.3.3), we have at once the following identifications :

Fd/SO(d) ≡ π0(Fd) = Hd , and Fd/SO(d− 1) ≡ π1(Fd) = Hd × ∂Hd .

Remark II.2.2.2 As we identified SO(d)(respectively SO(d − 1)

)with the elements of

PSO(1, d) fixing the first canonical vector e0 (respectively the vectors e0 , e1), we can iden-tify PSO(1, d − 1) with the elements of PSO(1, d) fixing the second canonical vector e1 .Considering the projection π′1 defined by π′1(β) := β1 = β(e1), we have then the followingidentifications, analogous to the above ones :

Fd/PSO(1, d− 1) ≡ π′1(Fd)

= Sittd , and Fd/SO(d− 1) ≡ π1

(Fd)≡ T 1Sittd ,

where Sittd :=ξ ∈ R1,d

∣∣ 〈ξ, ξ〉 = −1

is the one-sheeted unit hyperboloid (as Hd is the

upper sheet of the two-sheeted unit hyperboloid), known as the de Sitter space, and T 1Sittd isits unit tangent bundle. Note that T 1Hd and T 1Sittd differ only by their canonical projectionsonto Hd and Sittd respectively, which are the first one and the second one respectively. Inother words, the two first canonical vectors e0 , e1 have their roles exchanged between bothstructures. Moreover Proposition II.1.2 has the following analogue : the Minkowski pseudo-metric canonically induces a Lorentzian structure on Sittd, with PSO(1, d) and

(Fd, π′1

)as

isometry group and frame bundle respectively.

Proposition II.2.2.3 A sequence (qn) ⊂ Hd converges to η ∈ ∂Hd in the projective space

of R1,d (identifying qn with the line Rqn) if and only if〈qn, pη〉〈qn, p〉

goes to −1 for any p ∈ Hd,

or equivalently, for one p ∈ Hd. This implies that dist (p, qn) goes to infinity.


Proof Take a Lorentz frame β ∈ Fd such that β0 = p and β1 = pη , and write qn =d∑j=0

qjn βj . Then convergence in the projective space of R1,d holds if and only if q0n → +∞

and qjn/q0n → 1j=1 . Now, since |q0

n|2 −d∑j=1|qjn|2 = 1 , this holds if and only if q1

n/q0n → 1 .

As q0n = 〈qn, p〉 = ch [dist (p, qn)] and q1

n = −〈qn, pη〉 , the proof is complete.

Proposition II.2.2.4 Let η′, η be a geodesic of Hd, and let q ∈Hd \ η′, η. We have the following expression for the hyperbolic distancefrom the point q to the geodesic η′, η:

ch2[dist (q, η′, η)

]=

2

〈ηq, η′q〉. (II.2)

Moreover, the minimising geodesic from q to η′, η intersects the planeη′, η orthogonally. The intersection point is the orthogonal projectionof q on the geodesic η′, η in Hd.

Proof Given q ∈ Hd \ η′, η, it is straightforwardly verified that q′ := 〈ηq, η′q〉−1(ηq + η′q)is the pseudo-orthogonal projection of q on the vector plane generated by ηq, η′q, and that

q := (2〈ηq, η′q〉)−1/2(ηq + η′q) belongs to the geodesic η′, η. Recall that we have 〈ηq, η′q〉 > 0by Lemma I.2.1. By Proposition II.1.1, we have ch2[dist (q, q)] = 〈q, q〉2 = 2

/〈ηq, η′q〉.

Let p be any point on the geodesic η′, η. This implies p = x ηq + x′ η′q, with x, x′ > 0 and2xx′〈ηq, η′q〉 = 1 . Hence, we have :

ch[dist (q, p)]− ch[dist (q, q)] = 〈p, q〉 − 〈q, q〉 = x+ x′ −√

2/〈ηq, η′q〉 =[√

x−√x′]2≥ 0 ,

which shows that dist(q, q) does realize dist[q, η′, η

].

Applying the preceding to p instead of q , we get 2 = 〈ηp, η′p〉 . As we have ηp = 〈q, ηp〉 ηqand η′p = 〈q, η′p〉 η′q , we find indeed 2 = 〈ηp, η′p〉 = 〈q, ηp〉〈q, η′p〉〈ηq, η′q〉.

Finally, qε :=[1+2ε

√2/〈ηq, η′q〉+ε2

]−1/2(q+ε q) runs the geodesic q, q, so that a tangent

vector at q to this geodesic isdodεqε = q −

√2/〈ηq, η′q〉 q = q − q′ , and the proof is complete.


Remark II.2.2.5 (i) As shown in the above proof, we have also thealternative expression for (II.2), valid for any point p on the geodesicη′, η :

2

〈ηq, η′q〉= 〈q, ηp〉 × 〈q, η′p〉 . (II.3)

(ii) From the above proof, we verify that the orthogonal projection q ofq on the geodesic η′, η is closer than q to any point p on the geodesicη′, η, and strictly closer if q does not belong to the geodesic η′, η.We indeed have

ch[dist(p, q)

]= 〈p, q〉 =

⟨x ηq + x′ η′q ,

ηq+η′q√2〈ηq ,η′q〉

⟩=

√〈ηq, η′q〉

2(x+ x′) =

ch[dist(p, q)

]ch[dist(q, η′, η)

] .

We have then the following statement, similar to Proposition I.6.1,about the action of the subgroup Td−1 on the boundary ∂Hd, whichextends the Poincare coordinates to the boundary ∂Hd. Recall that

θ+u (e0 + e1) = (e0 + e1), and then that Tx,y(e0 + e1) = y (e0 + e1).

Proposition II.2.2.6 The Lie subgroup Td−1 of horizontal transla-tions acts transitively and properly on ∂Hd \ R+(e0 + e1) : for any lightray η ∈ ∂Hd \ R+(e0 + e1), there exists a unique θ+

u ∈ Td−1 such thatθ+u (e0 − e1) ∈ η . u is called the Poincare coordinate of η .

By convention, the Poincare coordinate of R+(e0 + e1) will be ∞ .

In other words, we have in the canonical basis e = (e0, . . . , ed), for aunique u ∈ Rd−1 :

ηe0= e0 +

( |u|2 − 1

|u|2 + 1

)e1 +

2

|u|2 + 1

d∑j=2

uj ej . (II.4)

Equivalently, the Poincare coordinate u of the light ray η in


∂Hd \ R+(e0 + e1) having coordinates proportional to (η0, . . . , ηd) in thecanonical basis (e0, . . . , ed) is given by :

uj =ηj

η0 − η1=−〈ηe0

, ej〉〈ηe0

, e0 + e1〉for 2 ≤ j ≤ d . (II.5)

Proof Formula (II.4) is merely given by subtracting the two firstcolumns of the matrix θ+

u (recall Definition I.4.1) and using PropositionII.2.2.1. It is then easily solved in a unique way, which yields Formula(II.5).

Remark II.2.2.7 Consider a sequence (qn) ⊂ Hd, having Poincare coordinates (xn, yn),and a light ray η ∈ ∂Hd, having Poincare coordinate u (as in Propositions I.6.1 and II.2.2.6).Then the sequence (qn) goes to the boundary point η (recall Proposition II.2.2.3) if and onlyif its Poincare coordinates (xn, yn) ∈ Rd−1 × R∗+ go to (u, 0) in the Euclidean topology of

Rd−1 ×R∗+ . Check this, as an exercise.

II.2.3 Harmonic conjugation

Proposition II.2.3.1 Consider two pairs η, η′ and η′′, η′′′ of distinct light rays in ∂Hd.Then the three following statements are equivalent.

(i) The intersection of the two vector planes defined by η, η′ and η′′, η′′′ is a line, andthese planes are perpendicular.

(ii) The two geodesics defined by the vector planes η, η′ and η′′, η′′′ intersect orthogonallyin Hd.

(iii)〈η, η′〉〈η′′, η′′′〉〈η, η′′′〉〈η′, η′′〉 = 4 and

〈η, η′′〉〈η′, η′′′〉〈η, η′′′〉〈η′, η′′〉 = 1 . (Note that these fractions make sense,

since by homogeneity we can think for each of these light rays of any spanning vector.)

Moreover, if these conditions are fulfilled, then

(iv) the intersection of the geodesics η, η′ and η′′, η′′′ is the point of Hd which belongs to√〈η, η′′〉〈η, η′′′〉 η′ +

√〈η′, η′′〉〈η′, η′′′〉 η (note that this is a well defined timelike direction ;

in particular it does not depend on the vectors we can choose in C to represent the light raysη, η′, η′′, η′′′) ;

(v) there exists g ∈ PSO(1, d) which exchanges at the same time η and η′, and η′′ and η′′′:g(η) = η′, g(η′) = η, g(η′′) = η′′′, g(η′′′) = η′′.


Remark II.2.3.2 (i) The conditions of Item (iii) in Proposition II.2.3.1 are not redundantfor d ≥ 3, as show the two following examples. Taking (in the canonical frame of R1,3) :η = R+(1, 1, 0, 0), η′ = R+(1,−1, 0, 0), η′′ = R+(1, 0, 1, 0), and

- either η′′′ = R+(1, 0, 0, 1), we get 2 for the first cross-ratio and 1 for the second ;

- or η′′′ = (2, 1, 0,√

3 ), we get 4 for the first cross-ratio and 3 for the second ;

and in both cases, the intersection of both planes reduces to 0.(ii) However, if d = 2, then the second condition entails the first one, while the recip-

rocal does not hold. Indeed, we can use a Lorentz frame such that η = R+(1, 1, 0), η′ =R+(1,−1, 0), and for η′′ = R+(1, cosα, sinα), η′′′ = R+(1, cosϕ, sinϕ), the second cross-ratio equals 1 if and only if tg 2(α/2) = tg 2(ϕ/2), i.e., if and only if α+ ϕ ∈ 2πZ, while thefirst cross-ratio equals 4 if and only if [tg (α/2)−3 tg (ϕ/2)][tg (α/2)+tg (ϕ/2)]cotg (ϕ/2) = 0 .

(iii) Note that by definition of the pseudo-norm on the exterior algebra :⟨ηp ∧ η′p, η′′p ∧ η′′′p

⟩= det

(〈ηp, η′′p〉〈ηp, η′′′p 〉〈η′p, η′′p〉〈η′p, η′′′p 〉

)= 〈ηp, η′′p〉〈η′p, η′′′p 〉 − 〈ηp, η′′′p 〉〈η′p, η′′p〉 ,

that we have :〈ηp, η′′p〉〈η′p, η′′′p 〉〈ηp, η′′′p 〉〈η′p, η′′p〉

= 1 ⇔⟨ηp ∧ η′p, η′′p ∧ η′′′p

⟩= 0 .

Definition II.2.3.3 Two pairs of distinct light rays η, η′ and η′′, η′′′ which satisfy theconditions of Proposition II.2.3.1, will be called harmonically conjugate.

In this case, the ideal quadrangle η, η′′, η′, η′′′ will be called a harmonic quadrangle.

Proof of Proposition II.2.3.1. We use Proposition II.2.2.1.

Suppose first that condition (i) holds, pick a non-null vector q in the intersection of thevector planes η, η′ and η′′, η′′′, and pick also some reference point p ∈ Hd. We haveq = αηp + βη′p = γη′′p + δη′′′p , and there exists some non-null vector aηp + bη′p orthogonal toη′′p , η

′′′p . This implies 0 = 〈αηp +βη′p, aηp + bη′p〉 = (αb+aβ) 〈ηp, η′p〉 , and then αb+aβ = 0 ,

so that we can suppose a = α , b = −β . Then we have 0 = 〈αηp − βη′p, η′′p〉 or equivalentlyα〈ηp, η′′p〉 = β〈η′p, η′′p〉, which implies αβ > 0 . Now, since 〈q, q〉 = 2αβ 〈ηp, η′p〉 > 0 , we can

suppose that q ∈ Hd (up to multiplying it by a scalar), so that it must belong to bothgeodesics of Hd defined by η, η′ and η′′, η′′′, and we can then take p = q .

The tangent at p to the line η, η′∩Hd is limit of chords joining p to (1 + 2ε)−1/2(p+ εηp),

so that limε→0

ε−1[(1 + 2ε)−1/2(p+ εηp)− p

]= ηp − p = pη spans this tangent. Since it is also

orthogonal to p , it must be collinear to αηp − βη′p , and then orthogonal to η′′p , η′′′p .

Thus (i)⇒ (ii) is proved.


Suppose then that condition (ii) holds, and denote by p the intersection of the geodesiclines defined by η, η′ and η′′, η′′′. As we just saw in the proof of (i) ⇒ (ii) above, thenon-null vector pη is tangent to the line η, η′∩Hd and orthogonal to p , and similarly pη′′

is tangent to the line η′′, η′′′′ ∩Hd. Hence pη belongs the plane η, η′ and is orthogonalto p, pη′′, hence to η′′, η′′′ . This proves (ii)⇒ (i), whence (i)⇔ (ii).

Moreover (still under hypothesis (ii)) p = αηp + βη′p = (α + β)p + (αpη + βpη′) implies

α + β = 1 and αpη + βpη′ = 0 , and then α = β = 12 and pη′ = −pη . And similarly

pη′′′ = −pη′′ . Whence ηp + η′p = 2p , and 〈ηp, η′p〉 = 〈η′′p , η′′′p 〉 = 2 .

We must also have 〈pη, pη′′〉 = 0 . Whence 〈ηp, η′′p〉 = 〈ηp, η′′′p 〉 = 〈η′p, η′′p〉 = 〈η′p, η′′′p 〉 = 1 .

These values obviously satisfy condition (iii), so that (ii)⇒ (iii) and (iv) is proved.

Furthermore, we can complete (p, pη, pη′′) into some Lorentz frame (p, pη, pη′′ , p3, . . . , pd),and consider the isomorphism g which fixes p, p3, . . . , pd and maps (pη, pη′′) on

(−pη,−pη′′) : it belongs to PSO(1, d) and is as in (v). This proves (ii)⇒ (v).

Suppose reciprocally that condition (iii) holds, fix some reference point p ∈ Hd, and

consider q0 :=√〈ηp, η′′p〉〈ηp, η′′′p 〉 η′p +

√〈η′p, η′′p〉〈η′p, η′′′p 〉 ηp , and

q :=q0√〈q0, q0〉

=(2 〈ηp, η′p〉

)− 12

[(〈ηp, η′′p〉〈ηp, η′′′p 〉〈η′p, η′′p〉〈η′p, η′′′p 〉

) 14

η′p +

(〈η′p, η′′p〉〈η′p, η′′′p 〉〈ηp, η′′p〉〈ηp, η′′′p 〉

) 14

ηp

]∈ Hd.

Similarly, set

q′ :=(2 〈η′′p , η′′′p 〉

)− 12

[( 〈ηp, η′′p〉〈η′p, η′′p〉〈ηp, η′′′p 〉〈η′p, η′′′p 〉

) 14

η′′′p +

(〈ηp, η′′′p 〉〈η′p, η′′′p 〉〈ηp, η′′p〉〈η′p, η′′p〉

) 14

η′′p

]∈ Hd.

We have then : 〈q, q′〉 =

√〈ηp, η′′p〉〈η′p, η′′′p 〉〈ηp, η′p〉〈η′′p , η′′′p 〉

+

√〈ηp, η′′′p 〉〈η′p, η′′p〉〈ηp, η′p〉〈η′′p , η′′′p 〉

= 1 by (iii).

Now, this means that q = q′ , so that q0 is indeed a non-null vector belonging to both planesη, η′ and η′′, η′′′.

Consider then u :=√〈η′p, η′′p〉〈η′p, η′′′p 〉 ηp −

√〈ηp, η′′p〉〈ηp, η′′′p 〉 η′p , which is clearly orthogo-

nal to q . By (iii) we have :

〈u, η′′p〉 =√〈η′p, η′′p〉〈η′p, η′′′p 〉〈ηp, η′′p〉 −

√〈ηp, η′′p〉〈ηp, η′′′p 〉〈η′p, η′′p〉 = 0 ,

which means that the non-null vector u of the plane η, η′ is orthogonal to the planeη′′, η′′′. This proves that (iii)⇒ (i), thereby concluding the proof.


Proposition II.2.3.4 Consider two pairs η, η′ and η′′, η′′′ of distinct light rays in ∂Hd,and their Poincare coordinates (recall Proposition II.2.2.6) u, u′, u′′, u′′′ respectively.

Then η, η′ and η′′, η′′′ are harmonically conjugate if and only if

|u− u′| × |u′′ − u′′′||u− u′′′| × |u′ − u′′| = 2 and

|u− u′′| × |u′ − u′′′||u− u′′′| × |u′ − u′′| = 1 .

If d = 2 (and then u, u′, u′′, u′′′ ∈ R ∪ ∞), this is equivalent to the more usual cross-ratiocondition :

[u, u′, u′′, u′′′] :=u′′ − uu′′ − u′ ×

u′′′ − u′u′′′ − u = −1 .

Proof The first claim is merely the transcription of Condition (iii) of Proposition II.2.3.1in terms of the Poincare coordinates, since by Definition I.4.1(ii) we have simply⟨

θ+u1

(e0 − e1), θ+u2

(e0 − e1)⟩

=⟨θ+u1−u2

(e0 − e1), (e0 − e1)⟩

= 2 |u1 − u2|2.

For d = 2 , this yields ε, ε′ ∈ ±1 such that

[u, u′, u′′, u′′′] = ε and (u′−u)×(u′′′−u′′)(u′′−u′)×(u′′′−u) = 2ε′ .

But writing (u′−u) = (u′−u′′′) + (u′′′−u) and (u′′′−u′′) = (u′′′−u′) + (u′−u′′), we get at

once(u′ − u)× (u′′′ − u′′)(u′′ − u′)× (u′′′ − u)

= [u, u′, u′′, u′′′]− 1 , so that the condition reduces to 2ε′ = ε−1 ,

which is clearly equivalent to ε = ε′ = −1 , or to ε = −1 as well.

Remark II.2.3.5 The Lorentz-Mobius group PSO(1, 2) acts transitively on the set of har-monic quadrangles of H2. Any harmonic quadrangle is isometric to the quadrangle

−1, 0, 1,∞ of the Poincare half-plane R×R∗+ .

Proof Of course, any isometry maps plainly a harmonic quadrangle on a harmonic quad-rangle. Reciprocally, by Remark II.3.4, a change of Lorentz frame, hence an isometry, mapsa given harmonic quadrangle onto another given one.

Let us however give an alternative proof that all harmonic quadrangles are isometric, con-

sidering the half-plane R × R∗+ and identifying an ideal vertex (i.e., a boundary point, or

light ray) with its Poincare coordinate by means of Proposition II.2.2.6. By using a first

homography (seen as an element of SL(2), recall Proposition I.1.5.1), we move one vertex to

∞. We move next the most left vertex (on the real line) to −1 by a horizontal translation,

and then the right neighbouring vertex of −1 to 0 by a dilatation (centred at −1). So far, we

have obtained the new quadrangle −1, 0, α,∞, harmonic too, so that the geodesics [−1, α]


and [0,∞] are orthogonal, which forces finally α = 1 : any harmonic quadrangle is indeed

isometric to −1, 0, 1,∞.

II.3 Flows and leaves

Recall that PSO(1, d) has a right action on the set Fd of Lorentzframes (recall Definition I.2.2 and Remark I.3.3). In particular, theright action of the subgroups (θt) and (θ+

u ) introduced in Definition I.4.1defines the two fundamental flows acting on Lorentz frames.

Definition II.3.1 The geodesic flow is the one-parameter group definedon Fd by :

β 7→ β θt , for any β ∈ Fd and t ∈ R . (II.6)

The horocyclic flow is the (d− 1)-parameters group defined on Fd by :

β 7→ β θ+u , for any β ∈ Fd and u ∈ e0, e1⊥ ≡ Rd−1. (II.7)

Proposition II.3.2 The projection π0(β θt), of the orbit of a Lorentzframe β under the action of the geodesic flow, is a geodesic of Hd.Precisely, this is the geodesic determined by the plane β0, β1, and wehave :

d

dt(β θt)0 = (β θt)1 , and dist

(β0, (β θt)0

)= |t| .

Proof The expression of θt yields at once : (β θt)0 = (ch t)β0+(sh t)β1 ,

and (β θt)1 = (sh t)β0 + (ch t)β1 =d

dt(β θt)0 . Finally, by Proposition

II.1.1 we have :

dist(β0 , (β θt)0

)= argch

[〈β0 , (β θt)0〉

]= argch [ch t] = |t| .

II.3. FLOWS AND LEAVES 51

Corollary II.3.3 The geodesic segment [p, p′] joining p, p′ ∈ Hd has length dist (p, p′), andis the unique minimizing curve joining p to p′.

Proof The first claim follows at once from Proposition II.3.2 : we have necessarily [p, p′] =(β θt)0

∣∣ 0 ≤ t ≤ dist (p, p′)

, for some Lorentz frame β . Then, if q belongs to some

minimizing curve joining p to p′ in Hd, by Remark II.2.2.5(ii) we must have :

dist (p, p′) = dist (p, q) + dist (q, p′) ≥ dist (p, q) + dist (q, p′) ≥ dist (p, p′) ,

which forces the equality dist (p, q) = dist (p, q) and then the belonging of q to the geodesic

η′, η, by Remark II.2.2.5(ii) again.

Note that the geodesic flow makes sense also at the level of line-elements of Hd : recalling the identification

(Hd × ∂Hd ≡ T 1Hd

)of

Section II.2.2, we can set π1(β) θt := π1(β θt).

Indeed, if π1(β) = π1(β′), then % := β−1β′ ∈ SO(d − 1), so that π1(β′ θt) = π1(β % θt) =π1(β θt %) = π1(β θt). Moreover, we have (βθt)0 + (βθt)1 = et (β0 + β1), for any real t .

Thus the action of the geodesic flow (θt) on the boundary componentof π1(β) ∈ Hd × ∂Hd is trivial, and θt moves the generic line-elementalong the geodesic it generates, by an algebraic hyperbolic distance t .

On the contrary, the horocycle flow does not make sense at the level ofline-elements (for d ≥ 3).

The geodesic determined by a given (p, pη) = J(p, η) = J π1(β) ∈ T 1Hd (using the

notation of Section II.2.2), with β ∈ Fd (determined up to the right action of SO(d− 1)), isparametrised by its real arc-length s as follows : s 7→ (ch s)β0 + (sh s)β1 .

In other words, any geodesic of Hd is the isometric image (under β) of the geodesic run bythe point having Poincare coordinates (0, es) (s being the arc-length). We can thus say thatit has Poincare coordinates (0, es) in the frame β (instead of the canonical frame).

Remark II.3.4 Example : For an appropriate β ∈ Fd, the image under β−1 of a givenharmonic quadrangle Q (recall Definition II.2.3.3) has Poincare coordinates −1, 0, 1,∞.Precisely, this means (recall Proposition II.2.2.6) that

β−1(Q) =R+θ

+−e2(e0 − e1),R+(e0 − e1),R+θ

+e2(e0 − e1),R+(e0 + e1)

.

Indeed, denoting the harmonic quadrangle by η, η′′, η′, η′′′, we can restrict to the 3-subspacecontaining it, and take : β0 to be the intersection of geodesics η, η′ and η′′, η′′′, β1 :=


Figure II.2: an horosphere H based at η = R+(e0 + e1)

(β0)η′′′ , β2 := (β0)η .(Note that writing −1, 0, 1 means actually (−1, 0), (0, 0), (1, 0) ; this

is a usual writing, since no ambiguity can occur, as long as it is clear that ideal points (i.e.,light rays) are considered.

)This extends Remark II.2.3.5 to d ≥ 2 .

Definition II.3.5 (i) Given η ∈ ∂Hd, a horosphere based at η is theintersection of Hd with an affine hyperplane (of R1,d) orthogonal to η .

Given a Lorentz frame β , let H(β) denote the horosphere based atR+(β0 + β1) and defined by the hyperplane (β0 + β1)

⊥ containing β0 :

H(β) := Hd ∩(β0 + (β0 + β1)

⊥).(ii) A horosphere H based at η determines the horoball H+, which isthe intersection of Hd with the closed affine halfspace of R1,d delimitedby H and containing η .

Proposition II.3.2 stated in particular that any geodesic is the projec-tion of an orbit of the geodesic flow. Analogously, the following proposi-tion states in particular that any horosphere is the projection of an orbit


of the horocyclic flow.

Proposition II.3.6 The horosphere through a Lorentz frame β is theprojection of its orbit under the action of the horocyclic flow : H(β) =

π0

(β θ+

Rd−1

). For any u ∈ Sd−2,

(β0 ,

dodε

(β θ+ε u)0

)belongs to T 1Hd.

Moreover, we have

H(β) = H(β′)⇐⇒(∃ x ∈ Rd−1, % ∈ SO(d− 1)

)β = β′ θ+

x % .

Proof By the Iwasawa decomposition, any point of Hd can be written(βθ+

u θy)

0for some positive y . Now for any positive y and u ∈ Rd−1

we have :⟨(βθ+

u θy)

0− β0 , β0 + β1

⟩=⟨θ+u θy e0 − e0 , e0 + e1

⟩= 〈θy e0, e0 + e1〉 − 1

= e−y − 1 .

By Definition II.3.5.(i), this shows that the current point(βθ+

u θy)

0∈ Hd

belongs to H(β) if and only if y = 0 , i.e., if and only if it belongs toπ0

(β θ+

Rd−1

). Then, for u ∈ Sd−2 we have :⟨

β0 ,dodε

(β θ+ε u)0

⟩=dodε〈e0, θ

+ε u e0〉 =

dodε

(1 + ε2|u|2/2) = 0 ,

and⟨dodε

(β θ+ε u)0 ,

dodε

(β θ+ε u)0

⟩=

⟨dodε

(θ+ε u e0),

dodε

(θ+ε u e0)

⟩= 〈u, u〉 = −1 .

Suppose H(β) = H(β′). According to Definition II.3.5, we have β0 =(β′ θ+

x )0 for some x ∈ Rd−1. Hence, up to changing β′ into β′ θ+−x , we

can suppose also β0 = β′0 . Set β′1 =:d∑j=1

λj βj , withd∑j=1

λ2j = 1 . We


must then have, for any u ∈ Rd−1 : (β θ+u )0 ∈ β0 + (β0 + β′1)

⊥, i.e.,

0 =⟨

12 |u|2(β0 + β1) + β(u) , β0 + β′1

⟩= 1

2|u|2(1− λ1)−d∑j=2

λj uj ,

whence λ1 = 1 and λ2 = . . . = λd = 0 . Hence, β′1 = β1 , and thenβ−1β′ ∈ SO(d− 1). The reciprocal is obvious, by Definition II.3.5.

Mixing the actions of both flows, we get the notion of stable leaf, asfollows. See Figure II.3 (for a two-dimensional picture in the half spaceof Poincare coordinates, according to Section II.5.2 below).

Definition II.3.7 For any light ray η , denote by Fd(η) the set of allframes β ∈ Fd pointing at η , i.e., such that β0 + β1 ∈ η , and callit the stable leaf associated to the light ray η . We shall also writeFd(R+(β0 + β1)

)=: Fd(β) the stable leaf containing β .

Proposition II.3.8 The flows act on each stable leaf Fd(η). Pre-cisely, Fd(β) is the orbit of β under the right action of the subgroupof PSO(1, d) generated by Ad ∪ SO(d− 1) :

Fd(β) =β % θt θ

+u

∣∣∣ % ∈ SO(d− 1), t ∈ R, u ∈ Rd−1

=β Tz %

∣∣∣Tz ∈ Ad, % ∈ SO(d− 1).

Proof We already noticed in Section I.4 that the lightlike vector (e0 + e1) ∈ R1,d is aneigenvector for each matrix θt θ

+u ∈ Ad. Owing to Definition II.3.1, this means exactly that

the flows act on each stable leaf Fd(η). The Iwasawa decomposition, applying Theorem I.7.2to γ−1, yields : PSO(1, d) =

% θt θ

+u

∣∣ % ∈ SO(d), t ∈ R, u ∈ Rd−1.

Since % ∈ SO(d) fixes (e0 +e1) if and only if it belongs to SO(d−1), this implies at once that

the subgroup of PSO(1, d) fixing the light ray R+(e0 + e1) ∈ ∂Hd is precisely% θt θ

+u

∣∣ % ∈SO(d − 1), t ∈ R, u ∈ Rd−1. Whence the first characterization. The second one follows at

once, by Formula (I.17) and Proposition I.4.3.


d

d

Figure II.3: four frames of the stable leaf Fd(η) ≡ Fd(β) and two horospheres of Hη

The following statement illustrates the instability of the geodesic flowin a simple way. The reason behind that relies actually on the commu-tation relations in the Lorentz-Mobius group.

Proposition II.3.9 For any β ∈ Fd and ε, t ∈ R , we have :

dist((β θt)0 , (β θ

+ε e2θt)0

)= argch

(1 + ε2

2 e−2t)

;

dist((β θt)0 , (βRε θt)0

)= argch

(1 + (1− cos ε) sh2t

),

where Rε denotes the planar rotation mapping e1 to (cos ε)e1−(sin ε)e2 .

Note that the former formula in Proposition II.3.9 above shows up astability of horospheres for positive t and an instability for negative t ;while the latter exhibits a systematic rotational instability of the geodesicflow. See Figures II.3 and II.4 (for two-dimensional pictures in the halfspace of Poincare coordinates, according to Section II.5.2 below).


Figure II.4: rotational instability of the geodesic flow

Proof We merely use Definition I.4.1 and Formula (II.1), according towhich we can clearly replace β by the canonical frame, so that we easilyobtain :

dist(

(β θt)0 , (β θ+ε e2θt)0

)= dist

(θt e0 , θ

+ε e2

θt e0

)= dist

((ch t)e0 + (sh t)e1 , (ch t+ ε2

2 e−t)e0 + (sh t+ ε2

2 e−t)e1 + e−te2

)= argch

(1 + ε2

2 e−2t)

;

anddist

((β θt)0 , (βRε θt)0

)= dist

(θt e0 , Rε θt e0

)= dist

((ch t)e0 + (sh t)e1 , (ch t)e0 + (sh t cos ε)e1 − (sh t sin ε)e2

)= argch

(1 + (1− cos ε) sh2t

).

II.4. PHYSICAL INTERPRETATIONS 57

II.4 Physical interpretations

II.4.1 Change of frame and relative velocities

In special relativity, space-time is represented by Minkowski’s space,seen as an affine space, no frame being canonically given

(thus the canon-

ical frame (e0, . . . , ed) we introduced for notational convenience has noparticular physical meaning

). Change of frames are obtained by ele-

ments of the Lorentz-Mobius group PSO(1, d).

For any V ∈ Rd such that |V | < 1, consider the boost in the directionof V defined by : BV := %V θargth |V | %

−1V , %V ∈ SO(d) being such that

%V e1 = V/|V |. In particular we have θt = B(th t) e1. Note that BV

does not depend on the choice of %V , since θt commutes with SO(d−1).Note also that for any %′ ∈ SO(d) we have %′BV = B%′V %

′.

The Cartan decomposition (recall Theorem I.7.1) can be written in thefollowing slightly different way : g = BV % , with uniquely determined% ∈ SO(d) and V ∈ Rd such that |V | < 1. We have ge0 = e0 +V√

1−|V |2.

Now if two frames β′, β′′ ∈ Fd are such that β′′ = β′BV % , we say thatV is the relative velocity of β′′ with respect to β′. We have then theaddition rule for velocities

(in which U ·V = −〈U, V 〉 denotes the usual

Euclidean inner product in Rd).

Proposition II.4.1.1 For any U, V ∈ Rd such that |U |, |V | < 1, wehave BU BV = BW % , with % ∈ SO(d) and

W =

(1 + U ·V

|U |2)U +

√1− |U |2

(V − U ·V

|U |2 U)

1 + U · V.

If two frames β, β′ ∈ Fd are such that β′ = βBU%′ and if V is the

relative velocity of β′′ ∈ Fd with respect to β′, then the relative velocityof β′′ with respect to β is the above W in which V is replaced by %′V .


Proof Setting r := argth |U | and λ := argth |V | , we first have

BU e0 = ch r(e0+U

), BU U = ch r

(|U |2e0+U

), BV e0 = chλ

(e0+V

).

Then

BUBV e0 = chλ ch r(e0 + U

)+ chλBU

(U ·V|U |2 U + V − U ·V

|U |2 U)

= chλ ch r((1 + U · V ) e0 + U

)+ chλ ch r U ·V|U |2 U + chλ

(V − U ·V

|U |2 U)

= chλ ch r(

(1 + U · V ) e0 + U + U ·V|U |2 U +

√1− |U |2

(V − U ·V

|U |2 U))

= chλ ch r (1 + U · V )(e0 +W

).

This shows that % := B−1W BU BV ∈ PSO(1, d) preserves the direction of

e0 , hence e0 , hence is a rotation.

Finally for β, β′, β′′ ∈ Fd such that β′′ = β′BV % and β′ = βBU%′, we

merely have β′′ = βBU%′BV % = βBUB%′V %

′% = βBW ′%′′.

II.4.2 Motion of particles

The motion of a particle of unit mass is represented by a C1 timelikepath ξs ∈ R1,d parametrized by its proper time, i.e., its arc length ;so that its (d+ 1)-velocity ξs belongs to Hd. If ξs is C2, ξs is itsacceleration.

Given a frame β ∈ Fd, ξs is represented by its coordinates(t(s), Xs

):

ξs = t(s) β0 +d∑j=1

Xjs βj . τ ≡ t(s) can be seen as an absolute time in

the frame β (note that this is an increasing function of s), and thenZjτ := Xj

t−1(τ) ∈ β⊥0 describes the path of the particle in the frame β .

The β-velocity Vτ = (V 1τ , . . . , V

dτ ) is defined by

V jτ :=

d

dτZjτ =

Xjs

t(s)=

Xjs√

1 + |Xs|2, with τ = t(s) .

II.4. PHYSICAL INTERPRETATIONS 59(Thus the relative velocity V of β′′ ∈ Fd with respect to β′ ∈ Fd (defined

in the above section II.4.1) is the β′-velocity of β′′0 .)

Equivalently, Xjs =

V jτ√

1− |Vτ |2. The β-energy is t(s) =

1√1− |Vτ |2

.

Thus we have ξs = 1√1−|Vτ |2

(β0 +

d∑j=1

V jτ βj

). Note that the magnitude

|Vτ | =√

d∑j=1

(V jτ

)2= th

[dist (β0, ξs)

]is always < 1 , 1 being the velocity

of light in the present normalisation.

Note also that Vτ is constant if and only if ξs is.

II.4.3 Geodesics

Recall from Section II.2.2 that Jξs(η) is the unit spacelike vector (i.e.,

of pseudo-norm −1) orthogonal to ξs such that Jξs(η) + ξs ∈ η . Thus

the equation ξs = Jξs(η) can be interpreted by saying that the motionξs is accelerated uniformly in the direction defined by η . On the otherhand, the equation (ξs, η) = (ξ0, η) θs means that the (d+ 1)-velocity ξsruns the half-geodesic determined by ξ0, η (recall Proposition II.3.2).

Actually the (d+1)-velocity ξs runs a geodesic line ending at η ∈ ∂Hd

if and only if the motion ξs is accelerated uniformly in the directiondefined by η . Precisely, we have the following.

Proposition II.4.3.1 Given η ∈ ∂Hd, a timelike C2 path (ξs) parame-trized by its proper time satisfies the differential equation ξs = Jξs(η)

for all s ≥ 0 if and only if it satisfies (ξs, η) = (ξ0, η) θs for all s ≥ 0 .

Proof Since Je0

(R+(e0+e1)

)= e1 (see Figure II.1), for any β ∈ Fd and

z ∈ Rd−1 × R∗+, applying β Tz we have J(βTz)0

(R+(β0 + β1)

)= (βTz)1 .


Choose β ∈ Fd such that β0 = ξ0 and β0 + β1 ∈ η , so that (ξ0, η) =π1(β). We can write ξs = (βTzs)0 for any s ≥ 0 and some zs ∈Rd−1 × R∗+, so that we have Jξs(η) = (βTzs)1 .

Hence the equation ξs = Jξs(η) is equivalent to dds(βTzs)0 = (βTzs)1 ,

and then to the first order equation : dds Tzs e0 = Tzs e1 , which

(owing

to Formula I.18 and to the initial condition z0 = (0, 1))

has a uniquesolution zs . Now by Proposition II.3.2 this unique solution is indeedTzs = θs = T(0,es) as wanted.

II.4.4 Horospheres

We have the following physical characterization of horospheres, whichlets appear the embedding of the hyperbolic space in R1,d as rather nat-ural.

Proposition II.4.4.1 Consider two particles ξs and ξ′s parametrizedby their proper times and accelerated uniformly in the direction definedby the same light ray η . Then the vector ξs − ξ′s converges in R1,d ass→∞ if and only if there is an horosphere H (unique if d = 2) basedat η containing the initial velocities ξ0 and ξ′0 .

(Then ξs and ξ′s belong

to the horosphere Hθs for any s > 0 .)

Proof According to Proposition II.4.3.1, we have (ξs, η) = (ξ0, η) θsand (ξ′s, η) = (ξ′0, η) θs for all s ≥ 0 . According to Proposition II.3.6,the initial velocities ξ0 and ξ′0 belong to a same horosphere based at ηif and only if they can be written π0(β) and π0(β θ

+u ) for some β ∈ Fd(η)

and u ∈ Rd−1, so that ξs = π0(βθs) and ξ′s = π0(βθ+u θs). Then we have

β−1(ξs − ξ′s)− β−1(ξ0 − ξ′0) =

∫ s

0

(θt e0 − θ+u θt e0) dt

= (1− θ+u )((sh s) e0 + (ch s− 1) e1

)= (e−s − 1)

(|u|22 (e0 + e1) + u

),

II.5. POINCARE BALL AND HALF-SPACE MODELS 61

which shows that (ξs−ξ′s) converges to ξ0−ξ′0− |u|2

2 (β0+β1)−β(u) ∈ R1,d

as s→∞ .

Reciprocally, if ξs = π0(βθr+s) and ξ′s = π0(βθ+u θs) belong to two distinct

horospheres of the stable leaf Fd(η), computing as above leads to

β−1(ξs − ξ′s)− β−1(ξ0 − ξ′0) = (θr − θ+u )((sh s) e0 + (ch s− 1) e1

)= (e−s − 1)

(|u|22 (e0 + e1) + u

)+(sh(s+ r)− sh s− sh r

)e0 +

(ch(s+ r)− ch s− ch r

)e1

=(er−1

2

)es (e0 + e1) +O(1) ,

which shows that (ξs − ξ′s) diverges, in the light direction η . Finally recall that the case of ξs and ξ′s evolving in two distinct stableleaves has been considered in Proposition II.3.9.

II.5 Poincare ball and half-space models

We present here the classical ball and upper half-space models for thehyperbolic space Hd. We shall use them essentially to draw pictures,namely Figures II.3, II.4, II.6, II.7, II.8, III.2, IV.1, IV.2, IV.3, IV.4,IV.5, IV.6, IV.7, IV.8, V.1, V.2, VIII.1, VIII.2, and X.1.

II.5.1 Stereographic projection and the ball model

The ball model is obtained by projecting Hd stereographically ontoBd (the unit open ball of Rd) from the vector −e0 . See Figure II.5.Precisely, denoting this stereographic projection by ψ, we have :

ψ((ch r) e0 + (sh r)u

)= sh r

1+ch r u = th(r/2)u for any r ≥ 0, u ∈ Sd−1.


Figure II.5: stereographic projection ψ(p) = b from Hd onto Bd

Note that ψ extends continuously to ∂Hd, by ψ(R+(e0 + u)

):= u .

Let us see how looks a geodesic(that is to say, the image under ψ of

a hyperbolic geodesic in Hd)

in this model.

Proposition II.5.1.1 In the ball model Bd, the geodesics appear eitheras diagonal segments, or as circles (intersected with Bd) orthogonal to∂Bd = Sd−1.

Proof As clearly ψ is invariant under the action of SO(d), it is enough to consider thegeodesic ended by e1 and u :=

((cosα) e1 + (sinα) e2

)(with cosα 6= 1 ; recall Remark

II.2.1.2). In Hd this geodesic is determined by the plane(ξ0 = λ+ µ , ξ1 = λ+ µ cosα , ξ2 = µ sinα , 0, . . . , 0)

∣∣λ, µ ∈ R,

and then is the image of R∗+ under the map :

λ 7−→(λ+ 1

4λ sin2(α/2),λ+ cosα

4λ sin2(α/2), sinα

4λ sin2(α/2), 0, . . . , 0

)∈ Hd.

II.5. POINCARE BALL AND HALF-SPACE MODELS 63

Note that the trivial case sinα = 0 yields a diagonal segment, so that we can restrict tosinα 6= 0. The analogue in Bd is thus got as the image of R∗+ under the map :

λ 7−→(

0 , ξ′1 = cosα+4λ2 sin2(α/2)

1+4λ(λ+1) sin2(α/2), ξ′2 = sinα

1+4λ(λ+1) sin2(α/2), 0, . . . , 0

)∈ Bd,

for which the equation (ξ′1− 1)2 + (ξ′2− tg 2(α/2))2 = tg 2(α/2) is directly verified. And this

equation is indeed that of the circle orthogonal to Sd−1 and containing e1 , u .

Let us then see how looks a horosphere(that is to say, the image

under ψ of a horosphere in Hd)

in this model.

Proposition II.5.1.2 In the ball model Bd, the horospheres appear ashyperspheres included in Bd and tangent to ∂Bd = Sd−1.

Proof We proceed as for Proposition II.5.1.1, using Definition II.3.5(i). Owing to the actionof SO(d), it is enough to consider the horosphere based at η = (e0 + e1) and containingp = (ch r) e0 + (sh r)u . In Hd this horosphere is determined by the hyperplane

(µ+ ch r , µ+ sh r , λ2 , . . . , λd)∣∣µ ∈ R , λ = (λ2, . . . , λd) ∈ Rd−1 ,

and then is the image of Rd−1 under the map :

λ 7−→(ch r + er|λ|2/2 , sh r + er|λ|2/2 ,λ2 , . . . , λd

)∈ Hd.

The analogue in Bd is thus got as the image of Rd−1 under the map :

λ 7−→(1 + ch r + er|λ|2/2

)−1(sh r + er|λ|2/2 ,λ2 , . . . , λd

)∈ Bd,

for which the equation(ξ′1− 1+er

2(1+ch r)

)2+ (ξ′2)2 + · · ·+ (ξ′d)

2 =(

1+e−r

2(1+ch r)

)2is directly verified.

And this equation is indeed that of a hypersphere tangent to Sd−1 at e1 .

II.5.2 Upper half-space model and the Poincare coordinates

As already mentioned just after Proposition I.6.1, the upper half-space model model is Rd−1 × R∗+. It is obtained by performing thefollowing inversion about e1 :

Bd 3 b 7−→ e1 + 2 |b− e1|−2(b− e1) =: z ≡ y e1 +x ≡ (x, y) ∈ Rd−1×R∗+ .


We have then |b− e1| × |z − e1| = 2 , and the above is equivalent to :

b = e1+2 |z−e1|−2(z−e1), or as well(

writing b =d∑j=1

bjej , x =d∑j=2

xjej

)to :

y =1− |b|2|b− e1|2

, and xj =2 bj

|b− e1|2for 2 ≤ j ≤ d .

Proposition II.5.2.1 (i) The above coordinates (x, y) are preciselythe Poincare coordinates of Proposition I.6.1.

(ii) In the upper half-space model Rd−1×R∗+, the geodesics appear eitheras vertical half-lines or as half-circles, both orthogonal to Rd−1, and thehorospheres appear either as horizontal hyperplanes or as hyperspherestangent to Rd−1.

Proof (i) Given p =d∑j=0

pjej ∈ Hd, we have the corresponding b =d∑j=1

bjej = ψ(p) =

(1 + p0)−1d∑j=1

pjej ∈ Bd, whence 1− |b|2 = 2/(1 + p0) and

|b− e1|2 = (1 + p0)−2[(p1 − p0 − 1)2 + (p0)2 − 1− (p1)2] = 2 (p0 − p1)/(1 + p0).

Therefore y = 1p0−p1 and xj = pj

p0−p1 : we recover (I.19).

(ii) It is enough to use Propositions II.5.1.1 and II.5.1.2 and to recall that the above inversion

about e1 is a conformal map, which maps orthogonal circles/lines to orthogonal circles/lines,

and tangent hyperspheres to tangent hyperspheres or to horizontal hyperplanes.

Remark II.5.2.2 The hyperbolic space can be seen as an abstractmetric space. Then the above models are parametrizations which dependon some non-canonical choices. Actually they depend on a referencepoint in the space and also (in the case of the half space) on a referencepoint on the boundary, which can be changed by an isometry.

We saw in Proposition II.3.2 that the geodesic flow drives any β ∈ Fd by a distance|t| . For any u ∈ Rd−1, we have the following horocyclic analogue : distH(β, β θ+

u ) ≡

II.6. A COMMUTATION RELATION 65

distH(β)

(β0, (β θ

+u )0

)= |u| , where distH(β)

(β0, (βθ

+u )0

)denotes the horocyclic distance, i.e.,

the hyperbolic length of the minimal curve which links β0 to (βθ+u )0 within the horocycle

H(β) (as depicted on Figure II.3). Whereas dist(β0, (βθ

+u )0

)= argch

[|u|2/2 + 1

].

This is seen at once by applying the isometry β−1, which maps β0 to e0 and H(β) to thehorizontal hyperplane having equation y = 1 in Poincare coordinates.

Hence given any p, q ∈ Hd, the horocyclic distance distH(p, q) = 4 sh2[dist(p, q)

/2]

does not

depend on the horosphere H containing them (which is also seen by mapping isometrically

the geodesic segment [p, q] to a vertical one). See Lemma X.3.2 for the horocyclic distance

from a point to a geodesic.

II.6 A commutation relation

We establish here the part we shall need, at several places, of thecommutation relation between an element Tx,y ∈ Ad and a rotation % ∈SO(d). We postpone the full commutation formula (we do not need) tothe appendix, see Section IX.2. According to the Iwasawa decomposition(Theorem I.7.2, applied to γ−1 instead of γ), there exist unique Tx′,y′ ∈Ad and %′ ∈ SO(d) such that

Tx,y % = %′ Tx′, y′ .

We have then Tx,y e0 = %′ Tx′, y′ e0 , and the following.

Theorem II.6.1 Denote by u(%) the Poincare coordinate of

R+%(e0 + e1)(u(%) =∞ if and only if % ∈ SO(d− 1)

). We have :

(i) u(%′) = y u(%) + x or equivalently (ii) u(%−1) =u(%′−1)− x′

y′.

Proof (ii) By Definition of u(%) and by Formula (II.5) we have

〈u(%), ej〉 =

⟨%(e0 + e1), ej

⟩⟨%(e0 + e1), e0 + e1

⟩ =〈% e1, ej〉

1 + 〈% e1, e1〉.


On the other hand, since Tx,y(e0 + e1) = y (e0 + e1) (recall Section I.4),for any v ∈ R1,d we have :⟨

%−1(e0 + e1), v⟩

= y⟨T−1x,y (e0 + e1), % v

⟩= y

⟨e0 + e1, Tx,y % v

⟩= y

⟨e0 + e1, %

′ Tx′, y′ v⟩

= y⟨%′−1(e0 + e1), Tx′, y′ v

⟩.

Hence we obtain :

u(%−1) = −d∑j=2

⟨u(%−1), ej

⟩ej = −

d∑j=2

⟨%−1(e0 + e1), ej

⟩⟨%−1(e0 + e1), e0 + e1

⟩ ej= −

d∑j=2

⟨%′−1(e0 + e1), Tx′, y′ ej

⟩⟨%′−1(e0 + e1), Tx′, y′(e0 + e1)

⟩ ej= −

d∑j=2

⟨%′−1(e0 + e1), x

′j(e0 + e1) + ej

⟩⟨%′−1(e0 + e1), y′ (e0 + e1)

⟩ ej =u(%′−1)− x′

y′.

(i) is deduced at once from (ii), applied to Tx′, y′ %−1 = %′−1 Tx, y .

A rotation % ∈ SO(d) such that % e1 6= −e1 plainly can be decom-posed in a unique way as % = Rα,σ % , with % ∈ SO(d − 1), α ∈ [0, π[ ,σ ∈ Sd−2 ⊂ Rd−1 ≡ e0, e1⊥, Rα,σ ∈ SO(d) denoting the planar ro-tation by angle α , in the plane oriented by (e1, σ) (for α 6= 0). ThusRα,σe1 = % e1 = (cosα) e1 + (sinα)σ.

Proposition II.6.2 For any % ∈ SO(d), using the unique decompo-sition % = Rα,σ % with % ∈ SO(d − 1), α ∈ [0, π] and σ ∈ Sd−2, wehave

u(%) = cotg (α/2)σ =% e1 + 〈% e1, e1〉e1

1 + 〈% e1, e1〉.

Then |e1 − % e1| = 2 sin(α/2) = 2√|u(%)|2+1

, or alternatively :∣∣u(%)∣∣ = cotg (α/2) =

∣∣u(%−1)∣∣ .


This yields the interpretation :

u(%)→∞ ⇐⇒ % e1 → e1 ⇐⇒ α→ 0 ⇐⇒ %→ SO(d− 1).

Proof By Definition of u(%) and by Formula (II.5) we have

〈u(%), ej〉 =

⟨%(e0 + e1), ej

⟩⟨%(e0 + e1), e0 + e1

⟩ =〈% e1, ej〉

1 + 〈% e1, e1〉=

(sinα) 〈σ, ej〉1− cosα

,

whence u(%) = cotg (α/2)σ =% e1 + 〈% e1, e1〉e1

1 + 〈% e1, e1〉. Therefore∣∣u(%)

∣∣ = cotg (α/2), and otherwise

|e1 − % e1|2 = (1− cosα)2 + (sinα)2 = 4 sin2(α/2) =4

cotg 2(α/2) + 1.

Finally, changing % into %−1 does not change α .

We shall need in Section V.3 two particular cases of the commutationrelation of Theorem II.6.1, for which it will be sufficient to restrict tod = 2, but with some more specification about the elements %′, x′, y′.We now gather what we shall use, in the following.

Proposition II.6.3 Restrict here to the case d = 2, and denote Rα

for Rα,−e2. Then we have the following commutation relations :

(i) for any α ∈ [0, π] and x ∈ R, there exists a unique α′ ∈ [0, π] suchthat cotg (α′/2) = cotg (α/2)− x and

θ+x Rα = Rα′ θ

+x′ θlog y′ , with y′ :=

1− cosα

1− cosα′, x′ :=

sinα′ − sinα

1− cosα′;

(ii) for any real r we have a unique αr ∈ [0, π] such that :

θrRπ2

= Rαr θ+sh r θlog ch r , with cotgαr = sh r .


Proof (i) By Theorem II.6.1 applied with y = 1 and Proposition II.6.2,we have :

θ+x Rα = Rα′ Tx′, y′ , with cotg (α′/2) = cotg (α/2)− x .

Then we have Tx′, y′ e0 = R−1α′ θ

+x e0 , whence by Formula (I.19) :

y′ = 〈Tx′, y′ e0 , e0 + e1〉−1 =⟨R−1α′ θ

+x e0 , e0 + e1

⟩−1

=⟨θ+x e0 , e0 +Rα′ e1

⟩−1=⟨θ+x e0 , e0 + (cosα′)e1 − (sinα′)e2

⟩−1

=(1 + x2

2 − (cosα′) x2

2 + (sinα′)x)−1

=(1 + x2 sin2(α′/2) + x sinα′

)−1

=(1 +

(cotg(α

′

2 )− cotg(α2 ))2

sin2(α′

2 )−(cotg(α

′

2 )− cotg(α2 ))

sinα′)−1

= sin2(α2 )/

sin2(α′

2 ) = 1−cosα1−cosα′

.

By Formula (I.19) again, we have similarly :

x′ = −y′ 〈Tx′, y′ e0 , e2〉 = −y′⟨R−1α′ θ

+x e0 , e2

⟩= − y′

⟨θ+x e0 ,Rα′e2

⟩= − y′

⟨θ+x e0 , (sinα

′)e1 + (cosα′)e2

⟩= y′ x

(x2 sinα′ + cosα′

)= y′ x

(sin(α

′

2 ) cos(α′

2 ) cotg(α2 )− cos2(α′

2 ) + cosα′)

= y′(cotg(α2 )− cotg(α

′

2 ))

sin(α′

2 )(

cos(α′

2 ) cotg(α2 )− sin(α′

2 ))

= sin−2(α′

2 )(

cos α2 sin α′

2 − cos α′

2 sin α2

)(cos α′

2 cos α2 − sin α′

2 sin α2

)= sin−2(α

′

2 ) sin(α′−α2 ) cos(α+α′

2 ) = sinα′−sinα1−cosα′

.

(ii) By Theorem II.6.1 applied with (x = 0 , y = er) and PropositionII.6.2, we have :

θrRπ2

= Rαr Tx′, y′ , with cotg (αr/2) = er cotg (π/4) = er ,

whence cotgαr = e2r−12 er = sh r .

Then we have Tx′, y′ e0 = R−1αrθr e0 , whence by Formula (I.19) :

y′ = 〈Tx′, y′ e0 , e0 + e1〉−1 =⟨θre0 , e0 +Rαre1

⟩−1


Figure II.6: proof of the commutation relation of Proposition II.6.3(i)

=⟨θre0 , e0 + (cosαr)e1 − (sinαr)e2

⟩−1=(ch r − (cosαr) sh r

)−1

=(ch r − th r sh r

)−1= ch r .

And by Formula (I.19) again, we have similarly :

x′ = −y′ 〈Tx′, y′ e0 , e2〉 = −y′⟨R−1αrθre0 , e2

⟩= −y′

⟨θre0 ,Rαre2

⟩= −y′

⟨θre0 , (sinαr)e1 + (cosαr)e2

⟩= y′ (sinαr) sh r = sh r .

Figure II.6 yields a visual proof of Proposition II.6.3(i) : it is enoughto verify that both isometries θ+

x Rα and Rα′ θ+x′ θlog y′ map (under their

right action) a given line-element ξ ∈ T 1H2 to some common line-element ξ′. We choose ξ based at the point of the half plane havingcoordinates (− sinα′, 1− cosα′) and pointing to the centre (0, 1) of thehorocycle based at 0. It is moved by θ+

x Rα : first to the line-elementbased at (− sinα, 1− cosα) and pointing to the centre of the horocycle


Figure II.7: proof of the commutation relation of Proposition II.6.3(ii)

and then to the vertical line-element ξ′ based at the same point ; andsimilarly by Rα′ θ

+x′ θlog y′ , to ξ′ as well.

Similarly, Figure II.7 yields a visual proof of Proposition II.6.3(ii) : wechoose now the line-element ξ based at the point having coordinates(−sh r , 1) and tangent to the (geodesic) circle centred at 0, which ismoved by θrRπ

2to the vertical line-element ξ′ based at (0, ch r), and

by Rαr θ+sh r θlog ch r as well.

II.7 Busemann function

Denote by Hη the family of all horospheres based at η ∈ ∂Hd (recallDefinition II.3.5 and Figure II.3).

For any p ∈ Hd, there exists a unique Hη(p) ∈ Hη such that p ∈ Hη(p).

Note that Hη(p) = (p+η⊥)∩Hd, so that Hη(p) = Hη(p′)⇔ p−p′ ∈ η⊥ .

II.7. BUSEMANN FUNCTION 71

For any β ∈ Fd, we have H(β) = HR+(β0+β1)(β0).

Theorem II.7.1 (i) For any η ∈ ∂Hd, H ∈ Hη , and any real t ,

the set Hθt :=

(β θt)0

∣∣ β ∈ Fd, β0 ∈ H , β0 + β1 ∈ η

belongs to Hη .

(ii) For any H,H ′ ∈ Hη , there exists a unique real BH,H ′ such thatHθBH,H′ = H ′.

Proof (i) Let us fix β ∈ Fd such that β0 ∈ H and β0 + β1 ∈ η , so that H = H(β), andlet us verify that Hθt = H(β θt). For any β ∈ Fd such that β0 ∈ H and β0 + β1 ∈ η , wehave also H = H(β), whence by Proposition II.3.6 : β = β θ+

x % , for some x ∈ Rd−1 and% ∈ SO(d− 1), whence β θt = β θt θ

+x e−t % , and then (β θt)0 = (β θt θ

+x e−t)0 ∈ H(β θt).

Reciprocally, if p ∈ H(β θt), then p = (β θt θ+u )0 = (β θ+

u et θt)0 for some u ∈ Rd−1, while

(β θ+u et)0 ∈ H(β) = H and (β θ+

u et)0 + (β θ+u et)1 = β

(θ+u et(e0 + e1)

)= β(e0 + e1) ∈ η .

(ii) Fix β, β′ ∈ Fd such that β0 ∈ H , β′0 ∈ H ′ and β0 + β1 ∈ η , β′0 + β′1 ∈ η , so thatH = H(β), H ′ = H(β′). By Definition II.3.7 we have Fd(β) = Fd(β′), so that PropositionII.3.8 yields t ∈ R , u ∈ Rd−1 and % ∈ SO(d − 1) such that β′ = β(% θt θ

+u ) = β θt θ

+u % ,

showing that H ′ = H(β′) = H(β θt) = Hθt . Moreover, if Hθs = Hθt , then as in the proofof (i) above, we can fix β ∈ Fd such that H(β θs) = H(β θt), whence by Proposition II.3.6 :β θs = β θt θ

+x % , for some x ∈ Rd−1 and % ∈ SO(d− 1), and then : θs−t = θ+

x % .

By uniqueness in Iwasawa decomposition (recall Theorem I.7.2), this implies s = t , i.e.,

uniqueness of the real t such that H ′ = Hθt .

Remark II.7.2 For any η ∈ ∂Hd, H ∈ Hη , p ∈ Hd, let pH ∈ Hd

be defined by : (pH , η) := (p, η) θBHη(p),H. Then pH is the orthogonal

projection (recall Proposition II.2.2.4) in Hd of p on H, and RpH isthe orthogonal projection in R1,d of Rp on the affine hyperplane pH+η⊥

defining H.

Proof Fix β ∈ Fd(η) such that π1(β) = (p, η), and then π1(β θt) = (pH , η), for t :=BHη(p),H . We have also Hη(p) = H(β) and Hη(pH) = H(β θt) = H. Then, for anyq = (β θt θ

+u )0 ∈ H , we have :

ch[dist (p, q)

]=⟨β(e0), β(θtθ

+u e0)

⟩= 〈θ−te0, θ

+u e0〉 = ch t+ |u|2

2 et ≥ ch t = ch[dist (p, pH)

].


d

d

Figure II.8: Busemann function Bη(p, q)

Finally, for any real λ we have :

〈λ p− λ et pH , η〉 = λ⟨β(e0 − etθte0), β(e0 + e1)

⟩= λ− λ et

⟨(ch t)e0 + (sh t)e1, e0 + e1

⟩= 0 ,

whence λ p− (λ et−1) pH ∈ pH +η⊥ ⊂ H+η⊥ : the orthogonal projection of λ p on H+η⊥

is (λ et − 1) pH ∈ RpH .

Using the notation introduced in the above theorem II.7.1(ii), we cannow define the Busemann function. See Figure II.8 (depicted in thePoincare upper half-space model of Section II.5.2).

Definition II.7.3 The Busemann function B·(·, ·) is defined by :

for any η ∈ ∂Hd and any p, q ∈ Hd, Bη(p, q) := BHη(p),Hη(q) .

Proposition II.7.4 For any η ∈ ∂Hd and any p, q ∈ Hd, we have

Bη(p, q) = log 〈p, ηq〉 = − log 〈ηp, q〉 .

II.8. NOTES AND COMMENTS 73

Proof Fix β, β′ ∈ Fd(η) such that β0 = p and β′0 = q . We have then H(β) = Hη(p) andH(β′) = Hη(q). Set t := Bη(p, q), so that H(β′) = H(β)θt = H(β θt), by Theorem II.7.1. By

Proposition II.3.6, there exist u ∈ Rd−1 and % ∈ SO(d− 1) such that β′ = β θt θ+u % , whence

q = β(θt θ+u e0), and then

〈q, ηp〉 =⟨β(θt θ

+u e0), β(e0 + e1)

⟩=⟨θ+u e0, θ−t(e0 + e1)

⟩= e−t

⟨e0, θ

+−u(e0 + e1)

⟩= e−t.

Finally, we have ηq = 〈p, ηq〉 ηp , whence 1 = 〈p, ηq〉〈q, ηp〉 .

Remark II.7.5 For any η ∈ ∂Hd and p ∈ Hd, (qk) ⊂ Hd, we have Bη(p, qk) → ∞ if and

only if the points of the horosphere Hη(qk) go uniformly to η in the projective space of R1,d

(in the sense of Proposition II.2.2.3).

Proof On one hand by Theorem II.7.1(ii) and RemarkII.7.2, we have

Bη(p, qk) = dist(p,Hη(qk)

)(as appears clearly on Figure II.8), and on the other hand by

Proposition II.7.4 we have Bη(p, qk)→∞ if and only if 〈p+pη, qk〉 → 0 . Hence Bη(p, qk)→∞ implies 〈pη, qk〉

/〈p, qk〉 → −1 , and then the statement follows directly from Proposition

II.2.2.3.

II.8 Notes and comments

The hyperbolic geometry can be presented in many ways. See [Gh],[Mr] and [Rac] for an historical account, starting with Riemann andLobatchevski.

The point of view we adopted, to derive all notions from the geometryof the Minkowski space, is not so common, though it gives some niceintrinsic formulas, such as (II.2). See also [Rac].

Classical hyperbolic geometry has been developed in two directions :

- in Lie group Theory, with more general groups (of rank one and beyond)and associated homogeneous spaces, see in particular [He] ;

- in differential geometry, with the study of manifolds with (in gener-ally non-constant) negative curvature and their generalizations, such asCAT(-1) spaces and Gromov hyperbolicity ; see [Bn], [CG], [Gr], [Pa].

Chapter III

Operators and Measures

The Casimir operator Ξ on PSO(1, d), i.e., the second order differen-tial operator associated with the Killing form, is the fundamental opera-tor of the theory. It commutes with Lie derivatives, induces the Laplaceoperator D on the affine group Ad, and the hyperbolic and spherical

Laplacians ∆ and ∆Sd−1

, for which we specify some decomposition for-mulas and analytical expressions.

After proving some fundamental properties of Haar measures on

groups, we determine the Haar measure of PSO(1, d) and show that itis indeed bi-invariant.

This chapter ends with the presentation of harmonic, Liouville, andvolume measures. We let them all derive from the Haar measure, as wellas their analytical expressions, and the fundamental properties of thehyperbolic Laplacian : covariance with isometries and self-adjointness.

III.1 Casimir operator on PSO(1, d)

Let us define the Casimir operator Ξ , by means of right Lie deriva-tives

(recall Formula (I.6)

)and of the pseudo-orthonormal basis of

so(1, d) exhibited in Proposition I.3.2.

75

76 CHAPTER III. OPERATORS AND MEASURES

Definition III.1.1 The Casimir operator on PSO(1, d) is the secondorder differential operator Ξ , defined on C2 functions on PSO(1, d) by :

Ξ :=d∑j=1

(LEj)2 −∑

1≤k<l≤d(LEkl)2 , (III.1)

where Ej , Ekl | 1 ≤ j ≤ d , 1 ≤ k < l ≤ d is the basis of so(1, d) metin Proposition I.3.2.

We show the invariance of the expression of Ξ under a change of basis,in the proof of Proposition III.1.2 below. Using the subalgebra τd−1 ofSection I.4, i.e., writing LEj = LEj − LE1j

for 2 ≤ j ≤ d , we haveimmediately the following alternative expression of Ξ :

Ξ = (LE1)2 +

d∑j=2

(LEj)2 −

d∑j=2

(LEj LE1j+ LE1j

LEj)−∑

2≤k<l≤d(LEkl)2 .

(III.2)

Note that∑

1≤k<l≤d(LEkl)2 and

∑2≤k<l≤d

(LEkl)2 are the Casimir operators

on SO(d) and SO(d− 1) respectively.

Proposition III.1.2 The Casimir operator Ξ commutes with any rightderivative LA , A ∈ so(1, d) : Ξ LA = LA Ξ , and also with the leftand right actions of PSO(1, d) : Ξ Lg = Lg Ξ and Ξ Rg = Rg Ξ ,for any g ∈ PSO(1, d).

Proof We show in 1) below the invariance of the expression of Ξ under a change of basis.In 2) below, we apply this to deduce the Ad(g)-invariance of Ξ . The latter will implythen directly the commutation with LA , and the commutation with Rg as well, since thecommutation with Lg is obvious from (I.7) and (III.1).

1) Let us verify first that for any basis (A`)1≤`≤d(d+1)/2 of so(1, d), and its dual basis

(A∗` ) defined by : K(A∗` , A`′) = 1`=`′ , we have Ξ =∑

1≤`,`′≤d(d+1)/2

K(A`, A`′)LA∗` LA∗`′ .

III.1. CASIMIR OPERATOR ON PSO(1, D) 77

Note that by Proposition I.3.2, this expression is correct for the particular basis Ej , Ekl.Then, if (A`) is another basis of so(1, d), we have A` =

∑m

Pm,`Am for some

P ∈ GL(d(d+1)2 ), whence A∗` =

∑m

P−1`,mA

∗m , since indeed

K

(∑m

P−1`,mA

∗m , A`′

)=∑m

P−1`,m

∑m′

Pm′,`′ K(A∗m , Am′) =∑m

P−1`,mPm,`′ = 1`=`′ .

Hence we have∑`,`′

K(A`, A`′)LA∗` LA∗`′ =∑`,`′

∑m,m′

∑n,n′

Pm,`Pm′,`′P−1`,nP

−1`′,n′ K(Am, Am′)LA∗n LA∗n′

=∑m,m′

∑n,n′

1m=n1m′=n′K(Am, Am′)LA∗n LA∗n′ =∑

1≤m,m′≤d(d+1)/2

K(Am, Am′)LA∗m LA∗m′ .

2) Let us then fix any g ∈ PSO(1, d), and apply 1) above, taking now A` := Ad(g)(A`).

By Formula (I.5), we have K(Ad(g)(A∗` ), A`) = K(A∗` , A`) = 1`=`′ , which shows that

A∗` = Ad(g)(A∗` ). Note also that for any C3 function F on PSO(1, d) and any A,A′ ∈so(1, d), we have for any h ∈ PSO(1, d) :

LA(F Ad(g)

)(h) =

dodεF(gh exp[εA]g−1

)=dodεF(ghg−1 exp[εgAg−1]

)=[LAd(g)(A)F

]Ad(g)(h),

whence

LALA′(F Ad(g)) = LA([LAd(g)(A′)F

]Ad(g)

)=[LAd(g)(A)LAd(g)(A′)F

]Ad(g).

Using Formula (I.5) again, we get therefore :

Ξ(F Ad(g)

)=

[∑`,`′

K(A`, A`′)LAd(g)(A∗` )LAd(g)(A∗`′ )F

]Ad(g)

=

[∑`,`′

K(Ad(g)(A`),Ad(g)(A`′)

)LA∗`LA∗`′F

]Ad(g)

=

[∑`,`′

K(A`, A`′)LA∗`LA∗`′F]Ad(g) = [ΞF ] Ad(g) ,

by 1) above, since (A`) is clearly another basis of so(1, d). Hence, taking g = exp(εA), by

Formula (I.4) we get Ξ(F exp[ε ad(A)]

)= [ΞF ] exp[ε ad(A)] , whence by differentiating

at 0 with respect to ε : Ξ(LAF ) = LA[ΞF ] , showing that the Casimir operator Ξ indeed

commutes with any LA , A ∈ so(1, d).


III.2 Laplace operator

We introduce here an important second-order operator, which inducesthe Laplacian on Hd, as we shall see in Section III.5.

Definition III.2.1 Let the Laplace operator D be the following secondorder differential operator, defined on PSO(1, d) :

D :=d∑j=2

(LEj)2 + (LE1

)2 + (1− d)LE1. (III.3)

The left invariance of the right Lie derivatives (recall Formula (I.7))entails that of their squares, and then of the Laplace operator D :

Lγ D = D Lγ , for any γ ∈ PSO(1, d).

Recall that we used the Iwasawa decomposition, in Theorem I.7.2,to define the canonical projection Iw from PSO(1, d) onto Ad. Thefollowing statement explains the relation between the Laplace operatorD and the Casimir operator Ξ .

Theorem III.2.2 The Casimir operator Ξ and the Laplace operatorD agree on functions on PSO(1, d) which are invariant under the rightaction of SO(d). In other words,

Ξ [f Iw] = D [f Iw] for any C2 function f on Ad.

Proof Fix any γ ∈ PSO(1, d). For 1 ≤ k, l ≤ d , we have first :

LEkl[f Iw](γ) =dodεf Iw

(γ exp[εEkl]

)=dodεf Iw(γ) = 0 .

Fix then 2 ≤ j ≤ d . Using Formula (I.4) we have

LE1jLEj [f Iw](γ) =

dodε

dodη

f Iw(γ exp[εE1j] exp[η Ej]

)

III.2. LAPLACE OPERATOR 79

=dodε

dodη

f Iw(γ exp

[ηAd

(exp[εE1j])(Ej

)])=

dodε

dodη

f Iw(γ exp

[η exp

[ε ad(E1j)

](Ej)

]).

Now, ad(E1j)(Ej) = ad(E1j)(Ej) = E1 and ad(E1j)(E1) = −Ej , sothat

exp[ε ad(E1j)](Ej) = E1j + (cos ε)Ej + (sin ε)E1 .

Hence,

LE1jLEj [f Iw](γ) =

dodε

dodη

f Iw(γ exp

[η[E1j+(cos ε)Ej+(sin ε)E1

]])=

dodε

(LE1j

+ (cos ε)LEj + (sin ε)LE1

)[f Iw](γ) = LE1

[f Iw](γ) .

By Formulas (III.2) and (III.3), we deduce at once :

Ξ [f Iw] =[(LE1

)2 +d∑j=2

(LEj)2 −

d∑j=2

LE1jLEj][f Iw] = D[f Iw] .

Note that this property lets D appear as canonical among all similaroperators, and allows to restrict it to the affine subgroup Ad. We expressnow the restrictions of Lie derivatives to Ad, using the parametrisationof the subgroup Ad by the Poincare half-space Rd−1 × R∗+ .

Proposition III.2.3 Restricted to Ad, Lie derivatives and the Laplaceoperator can be computed as follows : for any z = (x, y) ∈ Rd−1 × R∗+ ,and for 2 ≤ j ≤ d ,

LE1f(Tz) = y

∂

∂yf(Tz) ; LEjf(Tz) = y

∂

∂xjf(Tz) ; (III.4)

L′E1f(Tz) =

[y∂

∂y+

d∑k=2

xk∂

∂xk

]f(Tz) ; L′

Ejf(Tz) =

∂

∂xjf(Tz) ; (III.5)


D = y2

(∂2

∂y2+ ∆d−1

x

)+ (2− d) y

∂

∂y, (III.6)

∆d−1x denoting the Euclidean Laplacian of Rd−1 acting on the coordinate

x .

Proof For any C2 function f on Ad and any z = (x, y) ∈ Rd−1×R∗+ ,by Formula (I.17) we have :

LE1f(Tz) =

dodεf(TzT0,eε) =

dodεf(Tx,eεy) = y

∂

∂yf(Tz),

and

LEjf(Tz) =dodεf(TzTεej ,1) =

dodεf(Tx+yεej ,y) = y

∂

∂xjf(Tz).

The computation relative to left derivatives is very similar. The formula(III.6) relative to D follows at once from Formulas (III.4) and (III.3).

III.3 Haar measure of PSO(1, d)

A left Haar measure λ on a locally compact group G is a positiveRadon measure which is invariant under left translations :∫

G

LgF dλ =

∫G

F dλ for any g ∈ G and any test-function F on G .

Of course, the analogous notion relative to the right hand side is that ofa right Haar measure, and a Haar measure is both a left and right Haarmeasure.

For any measure λ on a group G , denote by λ its image under

the inverse map (γ 7→ γ−1) :

∫G

F dλ :=

∫G

F (γ−1)λ(dγ) , for any test-

function F on G .

HAAR MEASURE OF PSO(1, d) 81

A Lie group is said semisimple when its Lie algebra is. By PropositionI.3.2, this applies to PSO(1, d). A classical theorem

(see ([He] X,1,

Proposition 1.4) or ([Kn], Corollary 8.31))

states that any semisimple Liegroup is unimodular, which means that its left and right Haar measuresare the same. We give a short proof of this in Theorem III.3.4 below.Taking advantage of that, we exhibit in Theorem III.3.5 below a Haarmeasure for PSO(1, d), and actually provide an explicit expression of it,in terms of Iwasawa coordinates.

We first have the following well known result.

Proposition III.3.1 (i) The image λ of a left Haar measure λ bythe inverse map is a right Haar measure.

(ii) A left Haar measure on a locally compact group is unique, up toa multiplicative constant.

Proof Fix any compactly supported continuous function F on G and any g ∈ G .

(i) Let λ be any left Haar measure on G . Then we have :∫GF (γ g)λ(dγ) =

∫GF (γ−1 g)λ(dγ) =

∫GF ((g−1γ)−1)λ(dγ) =

∫GF (γ−1)λ(dγ) =

∫GFdλ .

(ii) Let λ1, λ2 be left Haar measures on a locally compact group G . Consider any

compactly supported continuous function f on G , such that λ1(f) :=

∫Gf dλ1 6= 0 , and for

any g ∈ G , set ϕf (g) := λ1(f)−1

∫GRgf dλ2 . Let us show that ϕf is continuous on G .

Indeed, fixing any g0 ∈ G , any ε > 0 , and a compact neighbourhood V of the unitelement, on the one hand, by uniform continuity of f , we have sup

G|Rgf −Rg0f | < ε if

g−1g0 remains in some neighbourhood Vε ⊂ V of the unit element, and on the other hand,again for g ∈ g0V

−1ε , |Rgf −Rg0f | is supported in the compact set K := Support(f)V g−1

0 ;

so that g ∈ g0V−1ε implies |ϕf (g)− ϕf (g0)| ≤ |λ1(f)|−1 λ2(K) ε .

Fix then any other compactly supported continuous function h on G , and note that(g, γ) 7→ f(g)h(γg) defines a compactly supported continuous function on G2. Using the leftinvariance properties of λ1, λ2 and (i), we have :

λ1(f)×∫Gh dλ2 =

∫Gf(g)

[ ∫Gh(γg)λ2(dγ)

]λ1(dg) =

∫G

[ ∫Gf(g)h(γg)λ1(dg)

]λ2(dγ)


=

∫G

[∫Gf(γ−1g)h(g)λ1(dg)

]λ2(dγ) =

∫Gh(g)

[∫Gf(γg)λ2(dγ)

]λ1(dg) = λ1(f)

∫Ghϕfdλ1

by definition of ϕf . Hence, we get

∫Gh dλ2 =

∫Ghϕf dλ1 .

As a consequence, if f1 is any other compactly supported continuous function on G

such that λ1(f1) 6= 0 , we must have 0 =

∫Gh (ϕf − ϕf1) dλ1 , for any compactly supported

continuous function h on G . Thus ϕf 6= ϕf1 is a λ1-negligible open set, and then must be

empty, by invariance of the non-null measure λ1. This means that ϕ := ϕf does not depend

on f . By definition of ϕf , we then get

∫Gf dλ2 = ϕ(e)

∫Gf dλ1 , which by linearity holds

for any compactly supported continuous function f on G , proving the proportionality.

Let us establish the existence of a Haar measure in the simpler caseof SO(d), together with a disintegration of it.

We already stressed (just before Proposition II.6.2) that a rotation% ∈ SO(d) such that % e1 6= −e1 can be decomposed in a unique wayas % = Rα,σ % , with % ∈ SO(d − 1), α ∈ [0, π[ , σ ∈ Sd−2 ⊂ Rd−1 ≡e0, e1⊥, Rα,σ ∈ SO(d) denoting the planar rotation by angle α , in theplane oriented by (e1, σ) (for α 6= 0). Thus

Rα,σe1 = % e1 = (cosα)e1 + (sinα)σ. Note that (α, σ) is determined byRα,σ , and that Rα,σ is determined by % e1 = Rα,σe1 =: σ ∈ Sd−1, sothat we can write Rα,σ ≡ Rσ as well.

Proposition III.3.2 There exists a unique Haar probability measured% on SO(d). It is invariant under the inverse map. Using the decom-position % = Rσ % , which identifies % ∈ SO(d) with (σ, %) ∈ Sd−1 ×SO(d − 1), we have d% = dσ d% , where dσ denotes the uniform prob-ability measure on the sphere Sd−1 and d% denotes the Haar probabilitymeasure on SO(d−1). Moreover, Rσ = Rα,σ can be written in a unique

way Rσ = exp

[α

d∑k=2

σkE1k

], with (α, σ) ∈ [0, π] × Sd−2, and we have

the disintegration : dσ = Γ(d/2)√π Γ(

(d−1)/2) (sinα)d−2dα dσ .


Proof The property is trivial on SO(2). Let us proceed by induction on d , showing that,if d% is a (% 7→ %−1)-invariant Haar measure on SO(d− 1), then d% := dσ d% is a (% 7→ %−1)-invariant Haar measure d% on SO(d). This will be sufficient, by Proposition III.3.1.

Fix any positive measurable function F = F (%) on SO(d), identified with F (σ, %) onSd−1 × SO(d− 1), and %0 = Rσ0 %0 ∈ SO(d). Set %1 = %1(%0, σ) := R−1

%0σ%0Rσ ∈ SO(d− 1),

so that %0 % = Rσ0 %0Rσ % = R%0σ %1(%0, σ) % . Then we have :∫F (%0 %) d% =

∫F(%0σ, %1(%0, σ) %

)dσ d% =

∫ [ ∫F(%0σ, %1(%0, σ) %

)d%

]dσ

=

∫F (%0σ, %) dσ d% =

∫F (σ, %) dσ d% =

∫F (%) d% ,

where we used successively the left Haar property of d% and the invariance of dσ under any%0 ∈ SO(d) (which holds by the very definition of dσ). Set then F (%) := F (%−1), and forany σ ∈ Sd−1, denote by σ := −2〈σ, e1〉e1 − σ its symmetrical with respect to Re1 . Byinvariance of d% under % 7→ %−1 and by invariance of dσ under % ∈ SO(d), we have :∫

F d% =

∫F (Rσ %−1) dσ d% =

∫F (%Rσ) dσ d% =

∫F (R%σ %) dσ d% =

∫F d% ,

showing the invariance of d% under % 7→ %−1 . By the above, we finally have :∫F (% %0) d% =

∫F (%−1%0) d% =

∫F (%−1

0 %) d% =

∫F (%) d% =

∫F (%) d% .

Hence d% is a Haar measure on SO(d), and a probability measure if d% is.

Moreover, we have exp

[α

d∑k=2

σkE1k

]e1 = (cosα) e1 − sinα

d∑k=2

σk ek , which indeed

equals σ for a unique (α, σ) ∈ [0, π]× Sd−2, implying then the required formula

Rσ = exp

[α

d∑k=2

σkE1k

]. Performing this change of variable

(using for example that dσ =∣∣Sd−2

∣∣−1|σ1|−1dσ2 . . . dσd−1 , and similarly for dσ), this implies also that

dσ = cd(sinα)d−2dα dσ , for a normalisation constant cd , given in the statement.

The beginning of Proposition III.3.2 is actually a particular case of ageneral statement. See ([He], X, 1). We present now a general proof forsemisimple Lie groups, both of unimodularity and invariance under theinverse map, in the spirit of ([He], X, 1).We begin with the following.


Lemma III.3.3 Let λ be a left Haar measure on a Lie group G. Then for any g ∈ G andany test-function F on G, we have :∫

GF (γ g)λ(dγ) =

∣∣det[Ad(g)]∣∣× ∫

GF (γ)λ(dγ) .

Proof Changing the variable γ ∈ G into gγg−1, we get :∫F (γ g)λ(dγ) =

∫F (g−1γ g)λ(dγ) =

∫F (γ)λ

(Ad(g)(dγ)

)=

∫F ×

∣∣ det Ad(g)∣∣ dλ .

Note finally that | det Ad(g)| does not depend on γ ∈ G , since Ad(g) operates here on G .

As a consequence of Lemma III.3.3, we see that a left Haar measure on a Lie group G

is also a right Haar measure if and only if∣∣ det Ad(g)

∣∣ = 1 for any g ∈ G. This entails the

following general result.

Theorem III.3.4 If the Lie group G is semisimple, then it is unimod-ular, and moreover, its Haar measures are invariant under the inversemap G 3 γ 7→ γ−1.

Proof (i) Fixing some basis of G and denoting by M and K the matrices of Ad(g) andof the Killing form respectively, the Ad-invariance of the Killing form K on G (recall SectionI.1.1) implies tMKM = K , whence (detM2 − 1) detK = 0 , whence | detM | = 1 by thehypothesis. Using Lemma III.3.3, this proves that G is unimodular.

(ii) By (i) above and by Proposition III.3.1, we know that any left Haar measure λon G is proportional to its image λ under the inverse map : λ = c λ . Otherwise, fix abasis (A1, . . . , An) of G, a small ε > 0 such that exp be a smooth diffeomorphism on n∑j=1

αjAj

∣∣∣ (α1, . . . , αn) ∈ B(0; ε) ⊂ Rn

, a continuous function F supported in this set,

and set F (α1, . . . , αn) := F(

exp[ n∑j=1

αjAj

]). Then we have λ(dγ) = κ dα1 . . . dαn at the

unit element 1 of G, for some κ > 0 , so that :

c

∫F dλ =

∫F (γ)λ(dγ) = κ

∫B(0;ε)

F (α1, . . . , αn) dα1 . . . dαn +O(ε)

= κ

∫B(0;ε)

F (−α1, . . . ,−αn) dα1. . . dαn +O(ε) =

∫F (γ−1)λ(dγ) +O(ε) =

∫F dλ+O(ε),


whence c = 1 +O(ε), and the conclusion follows by letting ε go to 0 .

This theorem applies in particular to PSO(1, d), which is semisimpleby Proposition I.3.2, hence unimodular.

We give in the following theorem an explicit expression for the Haarmeasure of PSO(1, d), in terms of its Iwasawa decomposition.

Theorem III.3.5 Let the measure λ be defined on PSO(1, d), usingthe Iwasawa decomposition (recall Theorem I.7.2), by :∫

PSO(1,d)

F dλ :=

∫Rd−1×R∗+×SO(d)

F (Tx,y %) y−d dx dy d% . (III.7)

The measure λ is a Haar measure on PSO(1, d), and moreover we haveλ = λ .

We begin with the affine subgroup Ad, which is not unimodular.

Lemma III.3.6 The Lie subgroup Ad = Tx,y | (x, y) ∈ Rd−1 × R∗+(recall Proposition I.4.3 and Formula (I.17)

)admits y−d dxdy as a left

Haar measure, and y−1 dxdy as a right Haar measure(dxdy denotes

the Lebesgue measure on Rd−1 × R∗+ ⊂ Rd).

Since these two measures are clearly non proportional (recall that d ≥ 2),Proposition III.3.1 ensures that Ad is not unimodular.

Proof Fix any compactly supported continuous function F on Ad,and any (x0, y0) ∈ Rd−1 × R∗+ . Using that Tx,y Tx′,y′ = Tx+yx′,yy′ , wehave at once :∫

Rd−1×R∗+LTx0,y0

F (Tx,y) y−d dxdy =

∫Rd−1×R∗+

F (Tx0+y0x,y0y) y−d dxdy

=

∫Rd−1×R∗+

F (Tx,y) y−d dxdy ,


whereas∫Rd−1×R∗+

RTx0,y0F (Tx,y) y

−1 dxdy =

∫Rd−1×R∗+

F (Tx+yx0,yy0) y−1 dxdy

=

∫Rd−1×R∗+

F (Tx,y) y−1 dxdy .

Lemma III.3.7 The Laplace operator D (recall Definition III.2.1) ofthe affine subgroup Ad is self-adjoint with respect to the left Haar mea-sure y−ddxdy (of Lemma III.3.6).

Proof For any test-function f, h on Ad, by Formula (III.6) we haveindeed : ∫

Rd−1×R∗+Df(Tx,y)× h(Tx,y) y

−ddxdy

=

∫ [∂2

∂y2+ ∆d−1

x

]f(Tx,y)× h(Tx,y) y

2−ddxdy+ (2− d)

∫∂

∂yf(Tx,y)× h(Tx,y) y

1−ddxdy

=

∫f(Tx,y)

[∂2

∂y2+∆d−1

x

](h(Tx,y) y

2−d)dxdy −(2−d)

∫f(Tx,y)×

∂

∂y

(h(Tx,y) y

1−d)dxdy

=

∫Rd−1×R∗+

f(Tx,y)×Dh(Tx,y) y−ddxdy .

Lemma III.3.8 For any test-function F on PSO(1, d) we have :∫PSO(1,d)

F dλ =


F (% Tx,y) y−1 dx dy d% .

Proof Using Proposition III.3.2, we have :∫PSO(1,d)

F dλ =


F (%−1 T−1x,y ) y−d dxdy d%


=


F (% T−xy−1, y−1) y−d dxdy d%

=


F (% Tx,y−1) y−1 dxdy d% =


F (% Tx,y) y−1 dxdy d% .

The proof of Theorem III.3.5 will moreover require the following in-finitesimal commutation relations.

Lemma III.3.9 For any (x, y) ∈ Rd−1 × R∗+ and for 2 ≤ j ≤ d , wehave :

Ad(T−1x,y )(E1j) =

(12y−

y2−

|x|22y

)Ej+y E1j+xj E1−

d∑k=2

xk Ejk+xjy

d∑k=2

xkEk ,

and for 2 ≤ i < j ≤ d : Ad(T−1x,y

)(Eij) = Eij −

xiyEj +

xjyEi .

Proof Denoted∑j=2

xjEj by E(x). By (I.13) and (I.14), we have :

ad[E(x)

](E1) = −E(x) , ad

[E(x)

](E1j) =

d∑k=2

xkEjk − xjE1 , ad[E(x)

](Eij) = xiEj − xjEi ,

whencead(E(x)

)2(E1j) = 2xj E(x)− |x|2Ej ,

and

exp[ad(E(−x))

](E1j) = E1j + xjE1 −

d∑k=2

xkEjk + xjE(x)− 12 |x|2Ej .

Similarly,ad(E1)(E1j) = Ej , ad(E1)2(E1j) = E1j ,

and then, by expanding the exponential :

exp[

log( 1y ) ad(E1)

](E1j) = 1

2 ( 1y − y)Ej + y E1j .

The first claim follows now from Formula (I.4). Similarly, for 2 ≤ i < j ≤ d :

ad(E(x)

)(Eij) = xiEj − xjEi , whence ad(E(x))2(Eij) = 0 ,


and thenAd(θ+

x )(Eij) = exp[ad(E(x))

](Eij) = Eij + xiEj − xjEi ,

whence finally, using that Eij commutes with θ1/y :

Ad(T−1x,y

)(Eij) = Ad

(θ+−x/y

)(Eij) = Eij −

xiyEj +

xjyEi .

Lemma III.3.10 Infinitesimal invariance implies global invariance : in

particular, if

∫LEijF dλ = 0 for any test-function F on PSO(1, d) and

any 1 ≤ i < j ≤ d , then

∫R%F dλ =

∫F dλ for any test-function F

and any % ∈ SO(d).

Proof The infinitesimal invariance implies that for any s ≥ 0 :

d

ds

∫F(γ exp[sEij]

)dλ(γ) =

∫LEijFs dλ = 0 , where

Fs(γ) := F(γ exp[sEij]

). This is sufficient, since the elements exp[sEij]

generate SO(d). End of the proof of Theorem III.3.5 By Lemmas III.3.6 and III.3.8, λ is clearly right-invariant under Ad. Thus, to prove that λ is a right Haar measure, it is enough to show itsright invariance under SO(d), and by Lemma III.3.10, to show its infinitesimal right invari-ance under so(d). Now, for any test-function F on PSO(1, d) and any 1 ≤ i < j ≤ d , by(I.8) we have :∫

LEijF dλ =

∫LEijF (% Tx,y)

dx dyy d% =

∫RTx,y(LEijF )(%) dx dyy d%

=

∫LAd(Tx,y)(Eij)(RTx,yF )(%) dx dyy d% =

∫L′Ad(Tx,y)(Eij)

(L%F )(Tx,y)dx dyy d% ,

by (I.10). Otherwise, by Lemma III.3.9 (applied to T−xy−1,y−1 = T−1x,y ), we have :

Ad(Tx,y)(E1j

)=(y2 − 1

2y −|x|22y

)Ej + 1

yE1j − xjy E1 + 1

y

d∑k=2

xkEjk +xjy

d∑k=2

xkEk ,

and for 2 ≤ i < j ≤ d :Ad(Tx,y)(Eij) = Eij + xiEj − xjEi .

THE SPHERICAL LAPLACIAN ∆Sd−1

89

By invariance of d% , the integrals containing a rotational derivative L′Eij vanish. Hence, wehave only to show that

0 =

∫ [(y2 − 1

2y −|x|22y

)L′Ej− xj

y L′E1+

d∑k=2

xkxjy L′Ek

](L%F )(Tx,y)

dx dyy d% ,

and

0 =

∫ [xi L′Ej − xj L

′Ei

](L%F )(Tx,y)

dx dyy d% .

Now, using Formulas (III.5) we have only to show that :

0 =

∫∂

∂xjL%F (Tx,y)

(y2 − 1

2y −|x|22y

)dx dyy d%−

∫∂

∂yL%F (Tx,y)xj

dx dyy d% ,

and that

0 =


∂

∂xjL%F (Tx,y)xi

dx dyy d%−


∂

∂xiL%F (Tx,y)xj

dx dyy d% .

Both formulas follow from obvious integrations by parts.

We have thus shown that λ is a right Haar measure on PSO(1, d). The proof is completedby applying Proposition I.3.2 and Theorem III.3.4.

Corollary III.3.11 The Lie derivatives are antisymmetric with respectto the Haar measure λ . The Casimir operator Ξ is self-adjoint withrespect to the Haar measure λ .

Proof For any test-functions F,G on PSO(1, d), and A ∈ so(1, d), by right invariance wehave : ∫

LAF ×G dλ =dodε

∫F(γ exp[εA]

)G(γ)λ(dγ)

=dodε

∫F (γ)G

(γ exp[−εA]

)λ(dγ) = −

∫F × LAGdλ ,

whence

∫(LA)2F ×Gdλ =

∫F × (LA)2Gdλ , and then the claim follows by Formula

(III.1).


III.4 The spherical Laplacian ∆Sd−1

Recall that we encountered the Casimir operator on SO(d) :

Ξ0 :=∑

1≤k<`≤d(LEk`)2 as a part of the Casimir operator Ξ on PSO(1, d),

in Formula (III.1). We use it now to derive the spherical Laplacian ofSd−1, by means of the projection % 7→ % e1 , from SO(d) onto Sd−1. Tothis aim, note that decomposing the generic rotation % ∈ SO(d) as inProposition III.3.2 :

% = Rσ % = exp

[d∑`=2

x`E1`

]% , with σ ∈ Sd−1, x ∈ Rd−1, % ∈ SO(d−1),

for any f ∈ C2(Sd−1) we have

LEk` e∗1f(%) =dodεf(Rσ exp

[εAd(%)[Ek`]

]e1

)= 0 for k, ` ≥ 2 .

Moreover it is clear that (as in the proof of Proposition III.3.2) :

% e1 = Rσ e1 = σ =(

cos |x|)e1 −

sin |x||x|

d∑`=2

x` e` .

Thus we see that Ξ0 e∗1f(%) =

d∑`=2

(LE1`)2 e∗1f(%) depends on % only

through % e1 ≡ σ ∈ Sd−1 ≡ SO(d)/SO(d − 1), thereby specifying anoperator on Sd−1. This allows the following definition.

Definition III.4.1 Setting e∗1f(%) := f(% e1) for any % ∈ SO(d), we

define the spherical Laplacian ∆Sd−1

by : for any f ∈ C2(Sd−1),

e∗1(∆Sd−1

f) := Ξ0 e∗1f =

d∑`=2

(LE1`)2 e∗1f .

THE SPHERICAL LAPLACIAN ∆Sd−1

91

This definition and the left-invariance of the right Lie derivatives implyat once the SO(d)-covariance of the spherical Laplacian : for any f ∈C2(Sd−1) and % ∈ SO(d), we have ∆Sd−1

(f %) =[∆Sd−1

f] % .

Note also that a straightforward adaptation of Corollary III.3.11 provesthat the Casimir operator Ξ0 is self-adjoint with respect to the Haarmeasure of SO(d).

The following proposition provides an expression of the sphericalLaplacian in Euclidean coordinates, and moreover useful and classicaldecompositions, which confirm in particular that it can as well be de-rived from the ambient Euclidean structure of Rd ⊃ Sd−1.

Proposition III.4.2 (i) Writing (σ1, . . . , σd) for the coordinates in Rd

of σ ∈ Sd−1 we have :

∆Sd−1

σ =∑

2≤j,k≤d(δjk − σjσk)

∂2

∂σj∂σk− (d− 1)

d∑k=2

σk∂

∂σk.

(ii) The Euclidean Laplacian ∆d of Rd can be decomposed in polarcoordinates (R , σ) ∈ R+ × Sd−1 according to :

∆d =∂2

∂R2+d− 1

R

∂

∂R+

1

R2∆Sd−1

σ .

(iii) The spherical Laplacian ∆Sd−1

can be decomposed in spherical polarcoordinates (α, σ) ∈ [0, π]× Sd−2

(writing σ ≡ (cosα, σ sinα) ∈ Sd−1 ⊂

Rd)

as :

∆Sd−1

σ =∂2

∂α2+ (d− 2) cotgα

∂

∂α+

1

sin2 α∆Sd−2

σ .

Proof (i) We have for 2 ≤ k, ` ≤ d :

LE1`σk = −LE1`

〈% e1, ek〉 = − dodε

⟨% exp[εE1`]e1, ek

⟩= 〈% e`, ek〉 .


Hence using the spherical coordinates (σ2, . . . , σd) we have :

LE1`e∗1f(%) = LE1`

f

( d∑k=1

σk ek

)=

d∑k=2

(LE1`σk)

∂

∂σkf(σ) =

d∑k=2

〈% e`, ek〉∂

∂σkf(σ) .

On the other hand,

(LE1`)2σk =

dodε

⟨% exp[εE1`]e`, ek

⟩= 〈% e1, ek〉 = 〈σ, ek〉 = −σk ,

and for 2 ≤ j, k ≤ d :

d∑`=2

(LE1`σj)(LE1`

σk) =

d∑`=2

〈% e`, ej〉〈% e`, ek〉 = −〈ej , ek〉 − 〈% e1, ej〉〈% e1, ek〉 = δjk − σj σk .

Thus we finally obtain the expression of (i) in the statement, since

d∑`=2

(LE1`)2 =

d∑`=2

[d∑

k=2

(LE1`σk)

∂

∂σk

]2

=∑

2≤j,k≤d(δjk − σj σk)

∂2

∂σj∂σk+

d∑k,`=2

(LE1`)2σk

∂

∂σk.

(ii) Starting from the Euclidean Laplacian ∆d, we change the variables (X1, . . . , Xd)

into the variables (R, σ2, . . . , σd), with σ` = X`/R and R =√X2

1 + · · ·+X2d : we have

∂

∂X`= σ`

∂

∂R+

d∑j=2

δj` − σj σ`R

∂

∂σj, and then

∂2

∂X2`

= σ2`

∂2

∂R2+

2

R

d∑j=2

(δj`− σ2` )σj

∂2

∂σj∂R+

1

R2

∑2≤j,k≤d

[δj`δjk+[σ2

` −δj`−δk`]σj σk] ∂2

∂σj∂σk

+1− σ2

`

R

∂

∂R− 1

R2

d∑j=2

[δj` − σ2

` +d∑

k=2

((1 + δj`)δk` − δjkσ2

` + (δ1` − δk`)σ2k

)]σj

∂

∂σj.

Summing over ` , the required expression (ii) follows now straightforwardly from (i) above.

(iii) We consider the additional variable r := R sinα , and change the variables (X1, r)

for the variables (R,α). We have R =√X2

1 + r2 and α = arccos(

X1√X2

1 +r2

), so that

∂

∂X1= cosα

∂

∂R− sinα

R

∂

∂αand

∂

∂X1= sinα

∂

∂R+

cosα

R

∂

∂α,

∂2

∂X21

= cos2 α∂2

∂R2+

sin2 α

R2

∂2

∂α2− sin(2α)

R

∂2

∂R∂α+

sin(2α)

R2

∂

∂α+

sin2 α

R

∂

∂R,

III.5. THE HYPERBOLIC LAPLACIAN ∆ 93

∂2

∂r2= sin2 α

∂2

∂R2+

cos2 α

R2

∂2

∂α2+

sin(2α)

R

∂2

∂R∂α− sin(2α)

R2

∂

∂α+

cos2 α

R

∂

∂R.

Hence by (ii) (applied to Rd−1) :

∆d =∂2

∂X21

+ ∆d−1 =∂2

∂X21

+∂2

∂r2+d− 2

r

∂

∂r+

1

r2∆Sd−2

σ

=∂2

∂R2 +d− 1

R∂

∂R+

1

R2

[∂2

∂α2+ (d− 2) cotgα

∂

∂α+

1

sinα2∆Sd−2

σ

],

which proves (iii), by using (ii) again.

III.5 The hyperbolic Laplacian ∆

We extend to functions the identification of Remark I.3.3 betweenPSO(1, d) and Fd, defining for any function h on PSO(1, d) the functionιh on Fd, by :

ιh(β) := h(β), for any β ∈ Fd . (III.8)

Remark III.5.1 The following statements about a function h on

PSO(1, d) are equivalent :

(i) h is invariant under the right action of SO(d) ;

(ii) the restriction g := h∣∣Ad of h to Ad is such that h = g Iw ;

(iii) there exists a function f on Hd such that ιh = f π0 .

Indeed, the equivalence between (i) and (ii) is obvious from the very definition of the Iwasawa

projection Iw , and the equivalence between (ii) and (iii) is easily seen, as follows : the

function h is invariant under the right action of SO(d) if and only if ιh on Fd is, and the

latter is clearly equivalent to (iii).

Theorem III.5.2 There exists a second order differential operator ∆on Hd, the so-called hyperbolic Laplacian , such that :


(i) ∆ is covariant with respect to hyperbolic isometries : we have

∆(f γ) = (∆f) γ for any γ ∈ PSO(1, d) ,

for any C2 function f on Hd, together with

(ii) (∆f) π0 = ι Ξ ι−1(f π0) = ι D ι−1(f π0) on Fd. (III.9)

Proof Write ι−1(f π0) =: h , so that by Remark III.5.1 h is rightSO(d)-invariant, and ιh = f π0 . Moreover by Theorem III.2.2 we haveDh = Ξh . Then using the right-invariance of Ξ we get :

Ξh = ΞR%h = R% Ξh , for any % ∈ SO(d) .

This proves the existence of ∆f satisfying (ii), again by Remark III.5.1.Then for any γ ∈ PSO(1, d), set fγ(p) := f(γ p) = f γ(p), and recall

Remark I.3.3 : γ(β) = γ β . Thus [fγ π0](β) =

= f(γ β0) = [f π0](γ(β)

)= ιh

(γ(β)

)= h

(γ(β)

)= h(γ β) = Lγh(β) ,

whence by (III.8) ι−1(fγ π0) = Lγh , and then the covariance property(i) of ∆ follows from the left-invariance of Ξ :

∆(fγ)(β0) = [∆(fγ) π0](β) = Ξ(ι−1(fγ π0)

)(β) = Ξ (Lγh)(β)

= Lγ Ξh(β) = Ξh(γ β) = Ξh(γ(β)

)= [(∆f) π0]

(γ(β)

)= (∆f)γ(β0) .

Let us now express the hyperbolic Laplacian in Poincare coordinates,

and also give an alternative definition of it, which underlines its analogywith the spherical Laplacian

(recall Definition III.4.1, where we used the

projection % 7→ % e1 from SO(d) onto Sd−1).

III.5. THE HYPERBOLIC LAPLACIAN ∆ 95

Corollary III.5.3 (i) In Poincare coordinates (recall PropositionsI.6.1 and III.2.3), the hyperbolic Laplacian is expressed by :

∆ = y2

(∂2

∂y2+ ∆d−1

x

)+ (2− d) y

∂

∂y. (III.10)

(ii) Using the projection γ 7→ γ e0 from PSO(1, d) onto Hd, let usset e∗0f(γ) := f(γ e0) for any f ∈ C2(Hd) and any γ ∈ PSO(1, d), orequivalently (using the notation of Remark I.3.3) :

e∗0f(β)

:= f(βe0

)= f π0(β) for any β ∈ Fd.

Then we have the following, which is equivalent to the definition formula(III.9) :

e∗0(∆f) = Ξ e∗0f = D e∗0f on PSO(1, d).

Proof (i) To obtain the expression (III.10), note that by the aboveproof of Theorem III.5.2 (with γ = Tz and the canonical basis as β) wehave :

(∆f)(Tz e0) = (∆f)Tz(e0) = ∆(fTz)(e0) = Ξ(ι−1(fTz π0)

)(1)

= Ξ (LTzh)(1) = LTz Ξh(1) = Ξh(Tz) = D h(Tz) .

Hence, noting that by the above we also have f(Tz e0) = h(Tz), we canapply Formula (III.6).

(ii) The equivalence of (III.10) with (III.9) is clear from (III.8), accord-ing to which we have e∗0 f = ι−1 f π0 on PSO(1, d).

We observe now that the Cartan decomposition of PSO(1, d) (recallTheorem I.7.1) induces naturally polar coordinates in Hd :

any p ∈ Hd can be written uniquely as p = (ch r) e0 + (sh r)φ ,

with coordinates (r, φ) ∈ R+ × Sd−1(and of course a singularity at the origin e0).

Indeed, the Cartan decomposition and Proposition I.6.1 yield directlyp = % θr %

′e0 = % θr e0 , whence φ = % e1 . We have then the following.


Proposition III.5.4 Using the polar coordinates (r, φ) ∈ R+ × Sd−1,

and the spherical Laplacian ∆Sd−1

φ of Section III.4 acting on the coordi-nate φ , we have :

∆ =∂2

∂r2+ (d− 1) coth r

∂

∂r+ (sh r)−2 ∆Sd−1

φ . (III.11)

Proof We use Formula (III.10) and Proposition I.6.1, and perform achange of variable.

Writing φ =d∑i=1

φi ei , this change between Poincare coordinates (y, x) = (y, x2, . . . , xd)

and polar coordinates (r, φ2, . . . , φd) is defined by : π0(Tx,y) = π0(Rφ θr) , where Rφ ∈ SO(d)

denotes the Euclidean rotation in the Euclidean plane e1, φ ⊂ Rd, mapping e1 to φ , thatis to say :

[y2 + |x|2 + 1

2y

]e0 +

[y2 + |x|2 − 1

2y

]e1 +

d∑j=2

xjyej = ch r e0 +

d∑j=1

φj sh r ej ,

meaning that

ch r =y2 + |x|2 + 1

2y;

xjy

= φj sh r for 2 ≤ j ≤ d , andy2 + |x|2 − 1

2y= φ1 sh r .

In particular, we have y−1 = ch r − φ1 sh r , and |x|2 = y2 (1− φ21) sh 2r .

Differentiating the above formulas for ch r and xj/y , we find :

∂r

∂xj= φj for 2 ≤ j ≤ d , ∂r

∂y= φ1 ch r − sh r ,

∂φi∂y

= −φi φ1/sh r ,

and∂φi∂xj

= (δij − φi φj)coth r − δij φ1 for 2 ≤ i, j ≤ d .

Hence

∂

∂y= [φ1 ch r − sh r]

∂

∂r− φ1

sh r

d∑i=2

φi∂

∂φi

and for 2 ≤ j ≤ d :

∂

∂xj= φj

∂

∂r+

d∑i=2

[(δij − φi φj)coth r − δij φ1

] ∂

∂φi.

III.6. HARMONIC, LIOUVILLE AND VOLUME MEASURES 97

This implies successively :

∂2

∂y2= [φ1 ch r − sh r]2

∂2

∂r2+

φ21

sh 2r

d∑2≤i,k≤d

φiφk∂2

∂φi∂φk− 2φ1(φ1coth r − 1)

d∑i=2

φi∂2

∂φi∂r

+[(1+φ2

1)sh (2r)

2−φ1ch (2r)+(1−φ2

1)coth r] ∂∂r

+[φ2

1coth 2r−φ1coth r+2φ2

1−1

sh 2r

] d∑i=2

φi∂

∂φi,

∂2

∂x2j

= φ2j

∂2

∂r2+ 2

d∑i=2

[(δij − φ2

j )coth r − δijφ1

]φi

∂2

∂φi∂r+[(1− φ2

j )coth r − φ1

] ∂∂r

+

d∑2≤i,k≤d

[(δijδik+[φ2

j−δij−δkj ]φiφk)coth 2r−(δij+δkj)(δik−φiφk)φ1coth r+δijδikφ21

] ∂2

∂φi∂φk

−d∑

k=2

[δjk − φ2

j

sh 2r−

d∑i=2

((2φ2

j − δjk − 1)coth 2r + (2δjk + 1)φ1coth r − δjk)]φk

∂

∂φk,

d∑j=2

∂2

∂x2j

= (1−φ21)∂2

∂r2+ 2φ1[φ1coth r− 1]

d∑i=2

φi∂2

∂φi∂r+[(d− 2 +φ2

1)coth r− (d− 1)φ1

] ∂∂r

+

d∑2≤i,k≤d

[(δik − [1 + φ2

1]φiφk)coth 2r − 2(δik − φiφk)φ1coth r + δikφ

21

] ∂2

∂φi∂φk

−d∑

k=2

[φ2

1

sh 2r+ 2φ2

1coth 2r − (d+ 1)φ1coth r + (d− 2)coth 2r + 1

]φk

∂

∂φk.

Substituting the above expressions in Formula (III.10), we find :

∆ =∂2

∂r2+ (d− 1) coth r

∂

∂r+

1

sh 2r

[ ∑2≤i,k≤d

(δik − φiφk)∂2

∂φi∂φk− (d− 1)

d∑k=2

φk∂

∂φk

],

which by Proposition III.4.2(i) yields the wanted formula (III.11).

III.6 Harmonic, Liouville and volume measures

The measures we introduce in this section play a fundamental rolein the theory, from the geometrical as well as from the probabilistic ordynamical points of view.


III.6.1 Harmonic measures and the Poisson kernel

Let us begin with the visual measures of the hyperbolic boundary :the harmonic measure µp must be understood as the measure of theboundary viewed from the point p .

Recall from Section II.2.2 that given any p ∈ Hd, Jp maps ∂Hd ontothe unit sphere of p⊥. Let σp⊥ denote the normalised Lebesgue measureon this sphere.

Definition III.6.1.1 The harmonic measure µp on ∂Hd is the imageof σp⊥ under J−1

p .

For short, using Proposition II.2.2.1, this means : µp(dη) = d pη .

We first have the following.

Lemma III.6.1.2 In Poincare coordinates, the harmonic measure µe0

is expressed by :

µe0(dη) =

2d−2 Γ(d/2)

πd/2(1 + |u|2

)1−ddu ,

where u ∈ Rd−1 is the Poincare coordinate of η ∈ ∂Hd (recall Proposi-tion II.2.2.6).

Proof By Formula (II.5), using Proposition II.2.2.1 we have :

uj =−〈ηe0

, ej〉〈ηe0

, e0 + e1〉, whence : |u|2 =

〈ηe0 , e0〉2−〈ηe0 , e1〉2〈ηe0 , e0+e1〉2 =

〈ηe0 , e0−e1〉〈ηe0 , e0+e1〉

,

and then

1 + |u|2 =2

〈ηe0, e0 + e1〉

=2

1 + 〈σ, e1〉, setting σ := (e0)η ∈ Sd−1.

We perform the change of variable σ 7→ u . Since ηe0= σ+e0 , we have :

uj =−〈σ, ej〉

1 + 〈σ, e1〉, or u =

σ + 〈σ, e1〉 e1

1 + 〈σ, e1〉= e1 +

σ − e1

1 + 〈σ, e1〉=: ϕ(σ) .


Then the differential dϕσ acts from the Euclidean (d−1)-space e0, σ⊥onto the Euclidean space Rd−1 by :

dϕσ = [1 + 〈σ, e1〉]−1 ×[1− σ − e1

1 + 〈σ, e1〉〈·, e1〉

]= [1 + 〈σ, e1〉]−1 × ` ,

with ` := 1− σ − e1

1 + 〈σ, e1〉〈·, e1〉 . Now, for any v ∈ e0, σ⊥, we have :

〈`(v), `(v)〉 = 〈v, v〉+ 2〈v, e1〉

1 + 〈σ, e1〉〈v, e1 − σ〉+

〈v, e1〉2[1 + 〈σ, e1〉]2

〈e1 − σ, e1 − σ〉 = 〈v, v〉 .

Hence, ` is an isometry from e0, σ⊥ onto e0, e1⊥ ≡ Rd−1. Thisproves that the Jacobian of ϕ at σ is [1 + 〈σ, e1〉]1−d, meaning byDefinition III.6.1.1 and by the above that :

2 πd/2

Γ(d/2)µe0

(dη) = |Sd−1|µe0(dη) = dσ = [1 + 〈σ, e1〉]d−1 du

=

(2

1 + |u|2)d−1

du .

Remark III.6.1.3 It is clear from Definition III.6.1.1 that for any hy-perbolic isometry g , since Jg(p) g = g Jp on ∂Hd, µg(p) is the imageof µp under g : µp g−1 = µg(p) .

This is the so-called geometric property of harmonic measures.

Moreover,µp∣∣ p ∈ Hd

is the unique family of probability measures on

∂Hd which enjoys this property.

Indeed, the geometric property determines the whole family from µe0 , and prescribes that

µe0 is SO(d)-invariant. Thus, µe0 Je0 must be an SO(d)-invariant probability measure on

Sd−1, hence indeed the normalized uniform measure.

We now obtain the expression of the harmonic measures in Poincarecoordinates, which brings out the Poisson kernel in a natural way, to-gether with a classical expression for it.


Proposition III.6.1.4 In Poincare coordinates, the harmonic measureµp (based at p ∈ Hd having Poincare coordinates (x, y) ∈ Rd−1 × R∗+,recall Proposition I.6.1) can be expressed as :

µp(dη) =2d−2 Γ(d/2)

πd/2

(y

y2 + |x− u|2)d−1

du , (III.12)

where u ∈ Rd−1 is the Poincare coordinate of η ∈ ∂Hd (recall Propo-sition II.2.2.6). The density which appears in Formula (III.12) is thePoisson kernel of Rd−1 × R∗+ .

Proof Lemma III.6.1.2 corresponds of course to p = e0 . To come now to the case ofan arbitrary p ∈ Hd, we use the geometric property quoted above, which tells that µp isthe image of µe0 under the isometry Tx,y . Hence, using Proposition II.2.2.6 and Lemma

III.6.1.2, we get for any test-function f on ∂Hd :∣∣Sd−1∣∣∫∂Hd

f dµp =∣∣Sd−1

∣∣∫ f(Tx,y(η))µe0(dη) =

∫Rd−1

f(Tx,y R+θ

+u (e0 − e1)

)[2

1+|u|2

]d−1du

=

∫f[Tx+yu,yR+(e0 − e1)

][ 2

1 + |u|2]d−1

du =

∫f[Tv,yR+(e0 − e1)

][ 2

1 + |v−xy |2

]d−1

d(vy

)

=

∫f[Tu,yR+(e0− e1)

][ 2 y

y2 + |u− x|2]d−1

du =

∫f[θ+uR+(e0− e1)

][ 2 y

y2 + |u− x|2]d−1

du .

It follows that all harmonic measures are equivalent. Precisely, wehave the following.

Proposition III.6.1.5 The harmonic measures µp are absolutely con-tinuous with respect to each other. More precisely, (using ηq of Propo-sition II.2.2.1) we have the relations : for any p, q ∈ Hd,

µq(dη) = 〈p, ηq〉d−1 µp(dη) = e(d−1)Bη(p,q) µp(dη) . (III.13)


Proof By Proposition III.6.1.4, we haveµe0(dη)

µp(dη)=

[y2 + |u− x|2y (1 + |u|2)

]d−1

, in Poincare coor-

dinates. Now, using Formulas (I.18) and (II.4), we have :

(1 + |u|2)× 〈p, ηe0〉 =[(y2 + |x|2 + 1)(|u|2 + 1)− (y2 + |x|2 − 1)(|u|2 − 1)− 4

d∑j=2

xjuj]/

(2y)

=[y2 + |x|2 + |u|2 + 2〈x, u〉

]/y =

y2 + |x− u|2y

.

This shows Formula (III.13) for q = e0 . The general case follows directly, since

µq(dη)

µp(dη)=

µe0(dη)

µp(dη)

/µe0(dη)

µq(dη)=

[〈p, ηe0〉〈q, ηe0〉

]d−1

= 〈p, ηq〉d−1 ;

indeed, since ηq and ηe0 must be collinear, we have immediately ηq = 〈q, ηe0〉−1 ηe0 , and

then : 〈p, ηq〉 = 〈p, ηe0〉/〈q, ηe0〉 . The alternative expression using the Busemann function is

merely due to Proposition II.7.4.

III.6.2 Liouville and volume measures

The Liouville and volume measures, on Fd and Hd respectively, arededuced from the Haar measure of PSO(1, d).

Definition III.6.2.1 The Liouville measure λ of Fd is the measureon Fd induced by the Haar measure λ of PSO(1, d), using the bijectionβ ↔ β of Remark I.3.3.

The volume measure of Hd is the image measure of the Liouville mea-sure under the canonical projection π0 (from Fd onto Hd). It will bedenoted by dp (so that dp = λ π−1

0 ).

Note that the Liouville measure λ on Fd is clearly invariant underthe left and right actions of PSO(1, d), and then in particular under thegeodesic and horocycle flows. This fundamental fact follows immediatelyfrom Definitions III.6.2.1 and II.3.1, and from Theorem III.3.5. With


Figure III.1: commutative diagram relating PSO(1, d), Fd, Hd, Ad, Iw, π0

Proposition II.1.2, this entails the invariance of the volume measure ofHd under hyperbolic isometries.

Note also that we have the commutative diagram III.1, in which β0 =Tx,y e0 has Poincare coordinates (x, y).

Proposition III.6.2.2 (i) The volume measure of Hd is expressed interms of the Poincare coordinates (x, y) ∈ Rd−1 × R∗+ as :

dp = y−d dx dy .

(ii) The Liouville measure λ of Fd disintegrates into λ = dp d% , where% ∈ SO(d) denotes the second Iwasawa coordinate, and d% denotes thenormalised Haar measure of SO(d) (recall Proposition III.3.2).

(iii) Using the polar coordinates (r, φ) of Proposition III.5.4, the volumemeasure of Hd is expressed as dp = (sh r)d−1 dr dφ .


Proof (i) The volume measure is indeed induced by the left Haar measure y−d dxdy onAd, by means of Poincare coordinates, due to Proposition I.6.1, Lemma III.3.6, and Formula(III.7). As a matter of fact, for any test-function f on Hd,

(extending π0 by setting

π0(β) := π0(β) = β0 = β(e0))

we have :∫Hd

f dp =

∫Fdf π0 dλ =

∫PSO(1,d)

f π0(γ)λ(dγ)

=


f π0(Tx,y %) y−d dx dy d% =

∫Rd−1×R∗+

f(Tx,y e0) y−d dx dy .

(ii) follows immediately, using the very definitions (III.7) and III.6.2.1 of the measures λand λ .

(iii) follows directly from the same change of variable as that of Proposition III.5.4.

Recall that we have : y−1 = ch r − φ1 sh r ,

∂r

∂xj= φj for 2 ≤ j ≤ d , ∂r

∂y= φ1 ch r − sh r ,

∂φi∂y

= −φi φ1/sh r ,

and∂φi∂xj

= (δij − φi φj) coth r − δij φ1 for 2 ≤ i, j ≤ d .

Hence the Jacobian J :=dr dφ2 . . . dφddy dx2 . . . dxd

is such that :

J−1 = (sh r)1−d det

φ1 ch r − sh r −φ1φ2 ....... −φ1φd

φ2 (1− φ22) ch r − φ1 sh r ....... −φ2φd ch r

... .... ....... ...φd −φ2φd ch r ....... (1− φ2

d) ch r − φ1 sh r

= (sh r)1−d det

φ1 ch r − sh r −φ1φ2 ....... −φ1φd

φ2 − φ2

φ1ch r (φ1 ch r − sh r) ch r − φ1 sh r ....... 0

... .... ....... ...

φd − φdφ1

ch r (φ1 ch r − sh r) 0 ....... ch r − φ1 sh r

= (sh r)1−d y1−d det

φ1 ch ρ− sh ρ −φ1φ2 ....... −φ1φd

φ2

φ1sh r 1 ....... 0

... .... ....... ...φdφ1

sh r 0 ....... 1

= (y sh r)1−d

(φ1 ch r − sh r + φ1φ2

φ2

φ1sh r + . . .+ φ1φd

φdφ1

sh r)

= (sh r)1−d y−d φ1 .


This shows that

dp = y−d dx dy = y−d J dr dφ2 . . . dφd = (sh r)d−1 φ−11 dr dφ2 . . . dφd ,

whence the result, since φ−11 dφ2 . . . dφd = dφ , as is easily computed, changing the Euclidean

variables (X1, . . . , Xd) for the variables (r, φ2, . . . , φd), as in the last part of the proof of

Proposition III.5.4 ; recall indeed that we have∂

∂Xi= φi

∂

∂r+

d∑j=2

δij − φi φjr

∂

∂φj, whence

the Jacobian

dr dφ2 . . . dφddX1 . . . dXd

= r1−d det

φ1 −φ1φ2 ....... −φ1φdφ2 1− φ2

2 ....... −φ2φd... .... ....... ...φd −φ2φd ....... 1− φ2

d

= r1−d φ1 ,

and then dX1 . . . dXd = rd−1 φ−11 dr dφ2 . . . dφd = rd−1 dr dφ .

Proposition III.6.2.3 The hyperbolic Laplacian ∆ (recall Section

III.5) is self-adjoint with respect to the volume measure dp . Similarly,

the spherical Laplacian ∆Sd−1

(recall Section III.4) is self-adjoint withrespect to the volume measure of Sd−1.

Proof The first statement is an immediate consequence of Formulas(III.6) and (III.10), Lemma III.3.7, and Proposition III.6.2.2(i). Thesecond statement is easily proved, either by induction on the dimen-sion (d− 1), using Propositions III.3.2 and III.4.2(iii), or from the firststatement, using Formula (III.11) and Proposition III.6.2.2(iii) (detailsare left as an exercise), or as well by using the self-adjointness of Ξ0

(as observed just after Definition III.4.1, as an adaptation of CorollaryIII.3.11), Definition III.4.1 and Proposition III.3.2, as follows :∫

Sd−1∆Sd−1

f(σ)× h(σ) dσ =

∫SO(d)

Ξ0(e∗1f)(%)× (e∗1h)(%) d%

=

∫SO(d)

(e∗1f)(%)× Ξ0(e∗1h)(%) d% =

∫Sd−1

f(σ)×∆Sd−1

h(σ) dσ .


We now specify the relation between the harmonic measures µp ofDefinition III.6.1.1 and the Liouville measure of Definition III.6.2.1.

Proposition III.6.2.4 (disintegration of the Liouville measure) Theprojection on the unit tangent bundle Hd × ∂Hd (recall Section II.2.2)of the Liouville measure λ can be written :

λ π−11 (dp, dη) = µp(dη) dp . (III.14)

In Poincare coordinates, this becomes :

λ π−11 (dp, dη) = 2d−2Γ(d2)π−d/2

(y2 + |x− u|2

)1−dy−1dx dy du. (III.15)

Proof We start from Proposition III.6.2.2(ii) : λ = dp d% .

A Lorentz frame β ∈ Fd is naturally decomposed into β0 ∈ Hd, β1 in thehyperplane β⊥0 , and the frame (β2, . . . , βd) ∈ β0, β1⊥. The right actionof SO(d) preserves β0 , and rotates β1 within the unit sphere of β⊥0 ,while the right action of SO(d − 1) rotates (β2, . . . , βd). Accordingly,d% is the product of the normalised volume measure dσβ⊥0 on the unit

sphere of β⊥0 and of the normalised Haar measure of SO(d − 1) (seeProposition III.3.2). Formula (III.14) follows then directly from the verydefinition III.6.1.1 of the harmonic measure. The expression (III.15) inPoincare coordinates results then immediately from (III.14) and fromPropositions III.6.1.4 and III.6.2.2(i).

Let us introduce now an alternative coordinate system on the unittangent bundle Hd × ∂Hd (recall Section II.2.2).

Definition III.6.2.5 For any fixed q ∈ Hd, and any (p, η) ∈ Hd×∂Hd,denote by η′ the other end of the geodesic determined by (p, η), and bys = sq(p, η) the algebraic hyperbolic distance to p , from the orthogonalprojection of q on the oriented geodesic (η′, η).


This provides global coordinates (η′, η, s) ∈ ∂Hd×∂Hd × R on theunit tangent bundle Hd × ∂Hd, where ∂Hd×∂Hd denotes the Cartesiansquare ∂Hd × ∂Hd without its diagonal.

Note that in this new coordinate system, the geodesic flow acts partic-ularly simply (recall Section II.3) : (η′, η, s) θt = (η′, η, s + t), for anys, t ∈ R.

We shall compute the expression of the Liouville measure in this newcoordinate system. We begin with the following sequel to PropositionIII.6.1.4.

Proposition III.6.2.6 Using Poincare coordinates (u′, u) of Proposi-tion II.2.2.6 for (η′, η), the Liouville measure on Hd×∂Hd can be writtenas :

λ π−11 (du′, du, ds) = 2d−2 Γ(d/2)π−d/2 |u′ − u|2(1−d) du′ du ds . (III.16)

Proof We start from the expression (III.15) in coordinates (y, x, u). We have to perform thechange of variable (y, x, u) 7→ (s, u′, u). We take q = e0 as the reference point of DefinitionIII.6.2.5.

We observe on Figure III.2 that the Pythagorean theorem provides the equation :

u′ = u+y2 + |x− u|2|x− u|2 (x− u). We use the polar angle ψ as an intermediate coordinate,

replacing momentarily the arc-length coordinate s . It satisfies :

x = 12 (u′ − u) cosψ + 1

2 (u+ u′) ; y = 12 |u′ − u| sinψ .

We then have

y2 + |x− u|2 = 14 |u′ − u|2 [sin2 ψ + (1 + cosψ)2] = 1

2 |u′ − u|2 (1 + cosψ) ,

whence (y2 + |x− u|2

)1−dy−1 = 2d |u′ − u|1−2d (1 + cosψ)1−d (sinψ)−1 .

On the other hand, by Remark II.1.4 we have : ds =|u′ − u| dψ

2 y=

dψ

sinψ.


y

ψ

x u´

Figure III.2: change of variable in the proof of Proposition III.6.2.6

Let us compute now the Jacobian Jd of the change of variables (x, y) 7→ (u′, ψ), for fixed u .We have

Jd = det

[∂x ∂y

∂u′ ∂ψ

]= det

12

(1+cosψ) 1d−1sinψ|u′−u| (u

′ − u)

t(u− u′) sinψ |u′ − u| cosψ

=: 2−d δd ,

where 1d−1 denotes the unit of GL(d− 1), and the vector (u′ − u) is seen as a column.

Expanding the determinant δd with respect to its first column yields the following recursion

formula : δd = (1 + cosψ) δd−1 + (1 + cosψ)d−2 sin2 ψ

|u′ − u| (u′ − u)2

1 , with δ1 = |u′ − u| cosψ ,

whence by induction :

δd = (1 + cosψ)d−1 δ1 + (1 + cosψ)d−2 |u′ − u| sin2 ψ = (1 + cosψ)d−1 |u′ − u| .Hence we get from the above :

λ π−11 (dp, dη) =

2d−2 Γ(d/2)

π−d/2(y2 + |x− u|2)1−d y−1 dx dy du

=2d−2 Γ(d/2)

π−d/2|u′ − u|2−2d dψ

sinψdu′du =

2d−2 Γ(d/2)

π−d/2|u′ − u|2−2d ds du′du .

We end with an intrinsic expression for Formula (III.16).


Proposition III.6.2.7 In the coordinate system of Definition III.6.2.5,depending on some reference point q ∈ Hd, the Liouville measure onHd × ∂Hd can be written as :

λ π−11 (dη′, dη, ds) = 2 πd/2 Γ(d/2)−1 〈η′q, ηq〉1−d µq(dη′)µq(dη) ds .

Proof Using the absolute continuity relation (III.13) : µq(dη) = 〈e0, ηq〉d−1 µe0(dη), andηq = 〈e0 , ηq〉 ηe0 (as already noticed at the end of the proof of Proposition III.6.1.5), we firstget :

〈η′q, ηq〉1−d µq(dη′)µq(dη) = 〈η′e0 , ηe0〉1−d µe0(dη′)µe0(dη) .

Hence we have only to consider the case q = e0 . We derive then this intrinsic expression fromthe preceding proposition III.6.2.6 and from Lemma III.6.1.2. We have indeed in Poincarecoordinates (recall Formula (II.4)) :

〈η′e0 , ηe0〉1−d µe0(dη′)µe0(dη)

du′ du=

⟨θ+u (e0 − e1)

1 + |u|2,θ+u′(e0 − e1)

1 + |u′|2⟩1−d

×[2d−2 Γ(d/2)π−d/2

]2[(1 + |u|2)(1 + |u′|2)]d−1

= 4d−2 Γ(d/2)2 π−d⟨θ+u−u′(e0 − e1), e0 − e1

⟩1−d= 2d−3 Γ(d/2)2 π−d |u− u′|2(1−d) .

III.7 Notes and comments

Casimir operators are usually presented in the more general contextof a semi-simple Lie group, and in the enveloping algebra formalism,under the form of the so-called Casimir element. See [Bi], [Ho], [Kn].

The invariance of the Haar measure is considerably simpler in twodimensions (d = 2), by using the PSL(2) model. See for example

([Ni],

Section 10.2).

The construction of the Laplacian via the Casimir operator trans-ported to the frame bundle has a certain analogy with the construc-tion of the Laplacian on a Riemanniann manifold as the projection ofthe Bochner horizontal Laplacian on the orthonormal frame bundle, see

III.7. NOTES AND COMMENTS 109

[KN]. Note however that the foliated Laplace operator D constructed inour section III.2 is not the horizontal Laplacian.

The disintegration of the Liouville measure (Proposition III.6.2.4) isclassical, see for example

([Ni], Section 8.1

). For a general Riemannian

manifold M, the Liouville measure will be defined on the unit framebundle T 1M. It is invariant under the geodesic flow, which describesthe evolution of the position and velocity of a free particle. It can beextended to the orthonormal frame bundle OM, by tensorization withthe Haar measure on SO(d−1). The geodesic flow can also be extended(via parallel transport) to the orthonormal frame bundle OM, whereit describes the motion of a solid. It preserves the extended Liouvillemeasure. See for example [KN].

Remark III.7.1 On a general Riemannian (or Lorentzian) manifold M, the metric (orpseudo-metric) is given infinitesimally, on the tangent bundle TM. In a given chart thiscorresponds to a symmetric matrix ((gjk)) = ((gjk(m)))m∈M (i.e., a quadratic form on TmM).

For example, in the particular case of Hd this amounts to the expression ds =

√|dx|2+dy2

y ofRemark II.1.4 for the hyperbolic norm of the line element given by the Poincare coordinates((x, y), (dx, dy)

). For R1,d and the polar coordinates (r, %, φ) (recall Section III.5), the expres-

sion of the Lorentzian pseudo-metric is easily seen to be : 〈dξ, dξ〉 = dr2−r2(dρ2+sh 2ρ |dφ|2

).

Then Formulas (III.10) and (III.11) for the hyperbolic Laplacian ∆(and similarly for the for-

mulas of Proposition III.4.2 relating to the spherical Laplacian ∆Sd−1

, and Formula (IX.6) forthe d’Alembertian 2 , see Section IX.3

)can be directly deduced from the general expression :

∣∣det((gjk(m)))∣∣−1/2 × ∂

∂mj

[∣∣det((gjk(m)))∣∣1/2 × gjk(m)

∂

∂mk

],

where ((gjk(m))) denotes the inverse matrix ((gjk(m)))−1.

Accordingly, the general expression for the volume measure (from which for example the

formulas of Proposition III.6.2.2 can be directly deduced) is∣∣det((gjk(m)))

∣∣1/2 dm .

Chapter IV

Kleinian groups

The hyperbolic space has infinite volume. To observe interesting dy-namical properties such as ergodicity, mixing, etc..., which will be stud-ied in Chapter V, we will consider periodic functions i.e., functions onspaces obtained by quotient. Such quotient space, associated with a dis-crete subgroup of isometries, is locally identical to the hyperbolic spacebut can have finite volume. The geodesic flows on such Riemann surfaces(when d = 2) are the simplest examples of mixing unstable flows definedby differential equations.

In this chapter we deal with the geometric theory of Kleinian groupsand their fundamental domains. We begin with the example of theparabolic tesselation of the hyperbolic plane by means of 2n-gons lim-ited by full geodesic lines, namely ideal 2n-gons. We discuss Dirichletpolyhedra and modular groups, with Γ(2), Γ(1), DΓ(1) and Γ(3) as mainexamples.

IV.1 Terminology

Definition IV.1.1 A Kleinian group is a discrete subgroup of

PSO(1, d). In other words, as a subset of PSO(1, d), a Kleinian group is a subgroup that

has no accumulation point.

111

112 CHAPTER IV. KLEINIAN GROUPS

Obviously, a Kleinian group is countable. Note also that a subgroupΓ is discrete if and only if the unit element 1 is not the limit of anyinjective sequence (γn) ⊂ Γ.

Indeed, if an injective sequence (γn) ⊂ Γ converges to some g ∈ PSO(1, d), then we can find

inductively a sequence (ϕn) ⊂ N∗ such that (γ−1n γn+ϕn) is injective, and it is clear that it

converges to 1 as n→∞ .

The main object of study associated to a Kleinian group Γ is its actionon Hd, and then its orbit space, that is, the space Γ\Hd := Γp | p ∈ Hdof the orbits of Hd under the (left) action of Γ. It is natural, in orderto visualize this orbit space, to consider subsets of Hd which essen-tially contain one and only one representative of each orbit. Precisely, afundamental domain for a Kleinian group Γ is a connected open subsetD ⊂ Hd with zero boundary volume, such that : γD ∩ γ′D = ∅ for anyγ 6= γ′ ∈ Γ and

⋃γ∈Γ

γD = Hd.

From the definition of a fundamental domain, the sequence of isometricimages of D under Γ covers Hd (up to a negligible subset) withoutoverlapping, drawing a paving, or tesselation of Hd.

Of course, there are fundamental domains D which are simpler thanothers, and more convenient for most purposes. Of particular interestare those which are polyhedral, i.e., bounded by a finite set of geodesichyperplanes.

Naturally, we say that a subset of Hd is convex if it contains the geodesicsegment linking any pair of its points. By the definition II.2.1.1 ofgeodesics and the identification of Hd with a part of projective Minkowskispace, this amounts to convexity within the projective Minkowski space.

Let us call convex polyhedron any convex closed subset P of Hd, pos-sibly unbounded, having its boundary made out of a finite number of(convex subsets of) geodesic hyperplanes of Hd, namely its sides. Withinthe projective Minkowski space, this can be identified with the intersec-

IV.2. DIRICHLET POLYHEDRA 113

tion of Hd with a finite number of half-spaces (containing a commontimelike ray).

A convex polyhedron is said to be ideal when all its vertices are ideal, meaning that they

belong to ∂Hd (i.e., are light rays).

A fundamental polyhedron for Γ will be a convex polyhedron, the inte-rior of which is a fundamental domain for Γ, and satisfying the followingadditional property : for each side S of P , there exists some γ ∈ Γ suchthat S = P ∩ γP .

A Kleinian group Γ admitting a fundamental polyhedron will be saidgeometrically finite.

Note that there are several other definitions of geometrical finiteness, essentially (but not

exactly) equivalent (see in particular [B1]), and that the present definition is more restrictive,

for d ≥ 4, than the ones considered in [B1] and [Rac]. For a more precise account, see([B1],

Section 5)

and([Rac], Section 12.3

). We shall consider only geometrically finite Kleinian

groups.

IV.2 Dirichlet polyhedra

We describe here an important way of obtaining fundamental poly-hedra for a Kleinian group Γ. Let us begin with a characterization ofKleinian groups. A subgroup Γ of PSO(1, d) is said to act discontinuously(or act properly discontinuously) on the hyperbolic space Hd if :

for any compact subset K ⊂ Hd, the set g ∈ Γ |K ∩ gK 6= ∅ is finite.

Lemma IV.2.1 A subgroup Γ of PSO(1, d) is Kleinian if and only ifit acts discontinuously on the hyperbolic space Hd.

In particular, any orbit Γp of an infinite Kleinian group Γ goes to in-finity in Hd. The torsion elements (i.e., the non-unit elements of finiteorder) of a Kleinian group are precisely its rotations.


Proof If the action of Γ were not discontinuous, we would have an injective sequence(gn) ⊂ Γ and a convergent sequence in Hd : xn → x ∈ Hd, such that gnxn → y ∈ Hd. Thuswe would have (gnx) bounded, (gn) equicontinuous and closed (since discrete) in Cc(Hd,Hd),and then compact by Ascoli Theorem, hence finite (since discrete), a contradiction. Inparticular, if an orbit Γp does not go to infinity, then there exists a compact set K0 suchthat K0 ∩Γp is infinite, and then applying the criterion with K = p∪K0 would yield thefiniteness of Γ. This criterion implies moreover that any rotation of Γ must generate a finitesubgroup i.e., be a torsion element, and conversely, by Theorem I.5.1 any torsion element canonly be a rotation.

Conversely, if Γ acts discontinuously on Hd, then any injective sequence (gn) ⊂ Γ convergingto the identity map, applied to any compact neighbourhood of any x ∈ Hd, would yield acontradiction, showing that Γ has to be discrete.

Note however that a Kleinian group generally does not act discontinuously on the bound-

ary ∂Hd.

For any p ∈ Hd which is not fixed by any element of a Kleinian groupΓ, the so-called Dirichlet polyhedron centred at p is

P(Γ, p) :=⋂γ∈Γ

q ∈ Hd

∣∣ dist(p, q) ≤ dist(γp, q). (IV.1)

Proposition IV.2.2 A Dirichlet polyhedron relative to Γ is a funda-mental polyhedron for Γ.

Proof For any p, p′ ∈ Hd, the half spaceq ∈ Hd

∣∣dist(p, q) ≤ dist(p′, q)

is convex, and

bounded by the geodesic hyperplaneq ∈ Hd

∣∣ dist(p, q) = dist(p′, q)

(the geodesic joiningtwo points of this hyperplane is contained in it), as is easily seen by choosing a half-spacePoincare model (i.e., Poincare coordinates in some frame, recall page 51) in which p, p′ havethe same vertical coordinate. Hence P(Γ, p) is clearly convex and closed, as an intersectionof geodesic hyperplanes of Hd.

Then for any R > 0 , by Lemma IV.2.1 the orbit Γp has finitely many points γ0p, . . . , γnp in

the compact set B(p, 2R), so that

[n⋂j=0

q ∈ Hd

∣∣∣dist(p, q) < dist(γjp, q)]∩B(p,R) is the

interior of P(Γ, p) ∩B(p,R) =

[n⋂j=0

q ∈ Hd

∣∣∣dist(p, q) ≤ dist(γjp, q)]∩B(p,R).

Hence, the interior of P(Γ, p) is⋂γ∈Γ

q ∈ Hd

∣∣ dist(p, q) < dist(γp, q)

, and P(Γ, p) is the

closure of its interior.

DIRICHLET POLYHEDRONS 115

In particular, P(Γ, p) has a locally finite number of sides, each of which is a convex includedin a hyperplane

q ∈ Hd

∣∣ dist(p, q) = dist(γp, q)

. And, since γ P(Γ, p) = P(Γ, γp) for any

γ ∈ Γ, we have P(Γ, p) ∩q ∈ Hd

∣∣ dist(p, q) = dist(γp, q)

= P(Γ, p) ∩ γP(Γ, p).

It is clear that for any γ ∈ Γ \ 1 and any q ∈ P(Γ, p), γq cannot be in the interior of P(Γ, p) :

this would indeed imply dist(p, γq) < dist(γp, γq) = dist(p, q) ≤ dist(γ−1p, q) = dist(p, γq).

Finally, for any q ∈ Hd, by discreteness, the minimal distance from q to the orbit Γp is

attained, for some γ0 : dist(Γp, q) = dist(γ0p, q), so that q ∈ γ0P(Γ, p).

IV.3 Parabolic tesselation by an ideal 2n-gon

When d = 2, Kleinian groups are often called Fuchsian. In thissection we describe in detail a planar example, exhibiting a particulartype of Fuchsian group. All the figures in the present chapter IV will usethe Poincare models B2 (disc) and R×R∗+ (upper half-plane) of SectionII.5.

Theorem IV.3.1 Let n ≥ 2, and consider 2n light rays, say η1, η2, . . . ,

η2n , arranged clockwise along ∂H2 (seen as a circle). Let P0 denote theideal 2n-gon : [η1, η2] ∪ . . . [η2n−1, η2n] ∪ [η2n, η1] , and for j ∈ 1, . . . , n,let ϕj denote the unique parabolic isometry fixing η2j and mappingη2j+1 onto η2j−1 (letting η2n+j ≡ ηj).

Then the group Ξn generated by ϕj | 1 ≤ j ≤ n is freely generated bythe ϕj’s, is Fuchsian, and admits the convex hull of P0 as a fundamentalpolygon. It has (n+ 1) inequivalent ideal vertices, and area 2(n− 1)π.

Note that P0 ≡ [η1, η2] ∪ . . . [η2n−1, η2n] ∪ [η2n, η1] (where each [ηj, ηj+1]denotes the geodesic from ηj to ηj+1) has no double point. We identifyP0 with the ordered set of its vertices : P0 ≡ η1, η2, . . . , η2n, and simi-larly for any (clockwise arranged) finite subset of the circle ∂H2, below.Note also that for each j ∈ 1, . . . , n, there exists indeed a unique


parabolic isometry ϕj(recall the third case of Proposition II.2.1.3 ; this

is also clear in the half-plane R × R∗+ , choosing Poincare coordinates(i.e., a frame, cf. page 51 and Proposition II.2.2.6) such that η2j ≡ ∞

).

Proof Observe that ϕj(P0) =η2j−1, ϕj(η2j+2), . . . , ϕj(η2j−1), η2j

is

arranged clockwise, and is contained in the arc of ∂H2 which joins η2j−1

to η2j and does not contain the points η2j+1, . . . , η2j−2 . Similarly,

ϕ−1j (P0) =

η2j, ϕ

−1j (η2j+1), . . . , ϕ

−1j (η2j−2), η2j+1

is arranged clockwise,

and is contained in the arc of ∂H2 which joins η2j to η2j+1 and does notcontain the points η2j+1, . . . , η2j−2 .(

See Figure IV.1 in the case n = 3, depicted in the disc model B2.)

Thus, the convex hulls of P0, ϕ1(P0), ϕ−11 (P0), . . . , ϕn(P0), ϕ

−1n (P0)

have pair-wise disjoint interiors, and their union constitutes the convexhull of the ideal 2n(2n− 1)-gon

P1 := P0 ∪ϕj(η2j+k+1), ϕ

−1j (η2j+k)

∣∣ 1 ≤ j ≤ n , 1 ≤ k ≤ 2n− 2

.

Then, the same procedure applies to each such 2n-gon ϕ±1j (P0) : for

1 ≤ ` < n , by means of the parabolic isometry ϕjϕj+`ϕ−1j fixing

ϕj(η2j+2`) and mapping ϕj(η2j+2`+1) onto ϕj(η2j+2`−1), and of the para-bolic isometry ϕ−1

j ϕj+`ϕj fixing ϕ−1j (η2j+2`) and mapping ϕ−1

j (η2j+2`+1)

onto ϕ−1j (η2j+2`−1).

Hence, taking the images of ϕj(P0) by ϕjϕ±1j+`ϕ

−1j , and then of ϕ−1

j (P0)

by ϕ−1j ϕ±1

j+`ϕj, for 1 ≤ ` < n and 1 ≤ j ≤ n , yields, in clockwise or-der, the following isometric 2n-gons, which have the interiors of theirconvex hulls pair-wise disjoint and disjoint from the convex hulls ofP0, ϕ1(P0), ϕ

−11 (P0), . . . , ϕn(P0), ϕ

−1n (P0) :

ϕjϕj+1(P0), ϕjϕ−1j+1(P0), . . . , ϕjϕ

−1j−1(P0), ϕ

2j(P0), ϕ

−2j (P0),

ϕ−1j ϕj+1(P0), . . . , ϕ

−1j ϕ−1

j−1(P0).

PARABOLIC TESSELATION BY AN IDEAL 2n-GONE 117

Figure IV.1: tesselation of H2 by an ideal hexagon P0, with parabolic pairings ϕ1, ϕ2, ϕ3


Going so on by induction, at the m-th step we obtain isometric 2n-gons

ϕε1

j1ϕε2

j1+j2. . . ϕεmj1+j2+···+jm(P0) , 1 ≤ jk ≤ n , εk = ±1 ,

indexed by the words of length m on the alphabet ϕ±11 , . . . , ϕ±1

n , suchthat the convex hulls of all isometric 2n-gons associated with the words oflength at most m have pair-wise disjoint interiors. And the union of thissame convex hulls, corresponding to all words of length at most m , is the(necessarily closed) convex hull of the ideal polygon Pm made of all ver-tices of all polygons ϕε1

j1ϕε2

j2. . . ϕεkjk (P0), 0 ≤ k ≤ m, j` ∈ Z/(2n)Z , εk =

±1 , arranged in clockwise order along ∂H2.

As a consequence of this construction, observe that for p in the convexhull of P0 and for any non-trivial reduced word ϕε1

j1ϕε2

j2. . . ϕεkjk , the point

ϕε1

j1ϕε2

j2. . . ϕεkjk (p) never belongs to the (closed) convex hull of P0 . This

implies immediately the following.

Lemma IV.3.2 The group Ξn is free, with rank n, and discrete.

Note that the even and the odd (ideal) vertices of P0 are intrinsicallydistinguished by the free group Ξn : each even vertex η2j is fixed bythe parabolic isometry ϕj . On the other hand, all odd vertices areequivalent under the group Ξn , and inequivalent to any even vertex,while two distinct even vertices of P0 are inequivalent. In particular,under the action of Ξn , there are exactly (n + 1) classes of vertices ofP0 , or of any Pm , or of

⋃m∈N

Pm .

Lemma IV.3.3 The hyperbolic area of the ideal polygon P0 equals2(n− 1)π .

Proof The ideal polygon P0 is obviously divided into 2(n − 1) idealtriangles, so that it is enough to verify that the area of any ideal triangleequals 2π . Now this is easily seen by using a Poincare half-plane model

PARABOLIC TESSELATION BY AN IDEAL 2n-GONE 119

in which one edge of the triangle is∞ . By using a dilatation (recall thisis a hyperbolic isometry), we have only to compute the area S of thegeodesic triangle 0, 1,∞ (half of the quadrangle P0 of Figure IV.3). Fi-

nally by Proposition III.6.2.2(i): S =

∫ 1

0

∫ ∞√x−x2

dyy2 dx = 2

∫ 1/2

0

dx√x−x2

= π.

The proof of Theorem IV.3.1 ends with the following.

Lemma IV.3.4 The increasing limit (as m → ∞) of the convex hullPm of the ideal polygon Pm , inductively defined above, covers the entirehyperbolic plane H2.

Proof Suppose that H2 \⋃m∈N

Pm contains a point p∞ . Pick p0 ∈

H2 ∩ ∂P0 , and consider the geodesic arc γ := [p0, p∞], which has finitelength ` := dist (p0, p∞). This arc cuts a sequence of fundamental do-mains P ′1 = ψ1(P0), . . . , P

′k = ψk(P0), . . . (we denote here also by P0 the

interior of the convex hull of P0), necessarily infinite since p∞ /∈ ⋃m∈N

Pm ,

and (by convexity of P0) such that j 6= k ⇒ P ′j ∩ P ′k = ∅ .

For any k ≥ 1, pick pk ∈ γ ∩ P ′k ∩ P ′k+1 , and for k ≥ 2 set p′k :=ψ−1k (pk−1), p

′′k := ψ−1

k (pk).

We thus have∑k≥1

dist (p′k, p′′k) ≤ ` < ∞ , with p′k, p

′′k ∈ ∂P0 , not on the

same edge of ∂P0 .

To any ideal vertex ηj , associate a horoball H+j based at ηj (recall

Definition II.3.5), in such a way that these horoballs be pair-wise disjoint.We need the following observation.

Lemma IV.3.5 Let Q := ∂P0 \2n⋃j=1

H+j . There exists ε > 0 such that

the hyperbolic distance from any p ∈ Q to ∂P0 \ Ep is ≥ ε, Ep denotingthe edge of ∂P0 containing p.


Proof It is enough to consider a given light ray η , and a Lorentzframe such that the Poincare coordinate of η is ∞ , the equation of thehoroball H+

η is y ≥ 1, and the equations of the edges of P0 startingat η are x = 0, x = a respectively, for some positive a . Then wehave to minimise dist

((0, y) ; (a, y′)

), for y ≤ 1 and y′ > 0 . Now, by

Formula (II.1) this equals argch[(a2 + y2 + |y′|2)/2yy′

], the minimum of

which is miny≤1

argch√

1 + (a/y)2 = argch√

1 + a2 > 0 .

End of the proof of Lemma IV.3.4 : The beginning of the proof andLemma IV.3.5 entail the existence of some k0 ≥ 1 such that for anyk ≥ k0 , the geodesic arc [p′k, p

′′k] is included in some horoball H+

jk.

Equivalently, the geodesic arc [pk−1, pk] must be included in the horoballψk(H+

jk). Since for any k the horoballs ψk+1(H+

j ) are pairwise disjoint,and as ψk(H+

jk) (containing pk) must be one of them, we must have

ψk(H+jk

) = ψk+1(H+jk+1

). This implies that the geodesic arc [pk0, p∞[ is

included in the horoball ψk0(H+

jk0). Now, this same geodesic arc must

intersect an infinite sequence of images of P0 , which are necessarilyof type ψiψk0

(P0), for some parabolic isometry ψ fixing the base ofψk0

(H+jk0

). Indeed, the subgroup of Ξn fixing a given ideal vertex is

conjugate to the subgroup fixing η2 , which reduces to the subgroup ϕZ1

generated by ϕ1 .

The desired contradiction is that no geodesic γ can intersect an infinitesequence of geodesics ψi(γ), provided ψ is parabolic fixing η and γ is ageodesic ending at η (η being not an end of γ). Indeed, by discretenessof the parabolic subgroup ψi | i ∈ Z and by compactness of ∂H2, thehalfspace of ∂H2 bounded by γ and not containing η can contain onlya finite number of ends of the geodesics ψi(γ) (as we clearly see on apicture). Remark IV.3.6 The Fuchsian group Ξn contains no rotation, but contains boosts.

Proof For the absence of rotation : consider a point z ∈ H2, fixed by some ϕ ∈ Ξn \ 1 .

IV.4. EXAMPLES OF MODULAR GROUPS 121

By Lemma IV.3.4, z = ψ(p) for some p ∈ P0 and ψ ∈ Ξn , so that ψ−1ϕψ fixes p and thenmust be trivial by the observation made just before Lemma IV.3.2, a contradiction.

As to the existence of boosts, consider any non-rotation isometry ψ . Taking the Poincarehalf-plane model R×R∗+ with ∞ fixed by ψ , it is easily seen that ψ becomes an element ofPSL(2) (recall Proposition I.1.5.1) which can be identified with a translation or a dilatation,so that in both cases, any sequence ψk(z0) goes to some boundary point z∞ ∈ ∂H2, as kgoes to infinity. We can apply this in particular to ϕ−1

2 ϕ1 , and to z0 = η4 . Noticing (seeFigure IV.1) that ϕ1(η4η5) ⊂ η1η2 and ϕ−1

2 (η1η2) ⊂ η4η5 , we obtain z∞ ∈ η4η5 which isfixed by ϕ−1

2 ϕ1 . Similarly, noticing that ϕ2(η2η3) ⊂ η3η4 and ϕ−11 (η3η4) ⊂ η2η3 , we obtain

z′∞ = limk→∞

(ϕ−11 ϕ2)k(η2) ∈ η2η3 which is fixed by ϕ−1

1 ϕ2 , hence by ϕ−12 ϕ1 too. This proves

that ϕ−12 ϕ1 is a boost (recall Theorem I.5.1).

Note that, whereas there are only finitely many non-isometric tesselations (by isometricregular convex polyhedra) of Euclidean space Rd , there are infinitely many non-isometrictesselations (by isometric regular convex polyhedra) of hyperbolic space Hd, as shown byTheorem IV.3.1 for d = 2. See also the simple example of triangle reflection groups

([Rac],

Section 7.2).

IV.4 Examples of modular groups

We restrict here to d = 2. By Proposition I.1.5.1, Fuchsian groups are isomorphic todiscrete subgroups of PSL(2). Thus, in the proofs of this section, we can and shall considerdiscrete subgroups of PSL(2). PSL(2) is in turn classically identified with the group H2 ofhomographies of the Poincare upper plane R×R∗+ ≡

z = x+

√−1 y ∈ C

∣∣x ∈ R, y ∈ R∗+

,

which is here the most convenient model for the hyperbolic plane H2, under the isomorphism :

PSL(2) 3 ±(a bc d

)7−→

(z 7→ az + b

cz + d

)∈ H2 .

The first example of a Fuchsian group is the full modular group Γ(1) := PSL(2,Z), madeof all classes of matrices having integer entries. The modular groups are the subgroups Γof Γ(1) having finite index [Γ(1) : Γ] . Most classical examples of modular groups are theprincipal congruence groups Γ(N), of classes of matrices congruent to the unit matrix moduloN ∈ N∗. They are normal in PSL(2,Z).

We shall take advantage below of Theorem I.5.1 yielding the classification of hyperbolicisometries

(belonging to PSO(1, 2) or equivalently to PSL(2)

)into rotations, boosts, and

parabolic elements.


IV.4.1 Plane tesselation by means of two parabolic isometries

Consider two parabolic isometries ϕ1, ϕ2 of the hyperbolic plane H2, with distinct fixedpoints p1, p2 respectively. We shall distinguish two cases. The first one, that we describe inthe following lemma, proves to be a particular case of Theorem IV.3.1 (with n = 2). On thecontrary the other case (considered in Proposition IV.4.1.3 below) will not be of this type.

Recall Definition II.2.3.3, restricted to d = 2 : a harmonic quadrangle is an ideal quadranglein H2, such that its four ideal vertices are determined by light rays which are harmonicallyconjugate, or equivalently, such that both geodesics joining opposite vertices intersect or-thogonally. Recall that the adjective “ideal” stresses the fact that the vertices are boundarypoints, i.e., located on the boundary ∂H2.

Proposition IV.4.1.1 The three following conditions are equivalent :

(i) the isometry ϕ1ϕ2 is parabolic (with fixed ideal point p) ;

(ii) the isometry ϕ2ϕ1 is parabolic (with fixed ideal point p′) ;

(iii) there exist two ideal points p, p′ ∈ ∂H2 \ p1, p2 such that ϕ2(p) = p′, ϕ1(p′) = p ,and the quadrangle P0 := p1, p, p2, p

′ is harmonic.

If these conditions hold, then the parabolic isometries ϕ1, ϕ2 are conjugate by means ofsome unique involutive isometry σ : (∃ ! σ ∈ PSO(1, 2) \ 1) ϕ1 = σ ϕ2 σ and σ2 = 1,

and also by means of two unique parabolic isometries g1, g2 (ϕ1 = gj ϕ2 g−1j for j = 1, 2).

Proof It is enough to consider the half-plane R × R∗+ , with p1 ≡ ∞ and p2 ≡ 0 . Thenϕ1 is an horizontal translation, by u ∈ R say, and up to conjugating by a dilatation, wecan suppose that u = 2 . Then we have necessarily ϕ2(z) = z

cz+1, for some c ∈ R∗. Thus,

ϕ1 ϕ2(z) = (2c+1)z+2cz+1 and ϕ2 ϕ1(z) = z+2

cz+2c+1, whence : ϕ1 ϕ2 is parabolic if and only if

c = −2 , and similarly for ϕ2 ϕ1 = ϕ−11 (ϕ1 ϕ2)ϕ1 .

In this case, the fixed points are p = 1 and p′ = −1 respectively, and we have on the onehand : ϕ1 = (z 7→ −1/z)ϕ2 (z 7→ −1/z), and on the other hand : ϕ1(−1) = 1, ϕ2(1) = −1 ,and clearly the quadrangle p1, p, p2, p

′ = ∞, 1, 0,−1 is harmonic.

Conversely, if ϕ2(p) = p′, ϕ1(p′) = p , and the quadrangle ∞, p, 0, p′ is harmonic, thenp = p′ + 2 , p− 2 = ϕ2(p) = p

cp+1, and p+ p′ = 0 , whence p = 1 and c = −2 .

Then, if σ2 = 1 6= σ , it is easily seen that σ = ±(

a −ba2+1b −a

), for real b 6= 0 and a ,

and if moreover ϕ1 σ = σ ϕ2 , then

(a 2a− b

a2+1b

2(a2+1)b − a

)= ε

(a −b

a2+1b − 2a 2b− a

)for some

ε = ±1 . This forces a = 0 , and then ε = 1 , whence b2 = 1 , that is, σ = (z 7→ −1/z) isindeed the unique solution.


Finally, if ϕ1 g = g ϕ2 for some parabolic isometry g , then g = ±(u vw 2− u

), for real

u, v, w such that (u − 1)2 = −vw , and

(u 2u+ vw 2w − u+ 2

)= ε

(u v

w − 2u 2− u− 2v

)for

some ε = ±1 . This forces u = 0 , and then ε = 1 , whence w = −v and then v2 = 1 , thatis, only the two following are solutions : g(z) = 1

2−z or g(z) = −1z+2

.

Remark IV.4.1.2 Any involutive (non-trivial) isometry of H2 has a unique fixed point(belonging to H2), and all involutive isometries of H2 (the identical map excepted) constitutea conjugacy class in the Lorentz-Mobius isometry group PSO(1, 2).

Proof By Theorem I.5.1, an involutive isometry of Hd cannot be parabolic nor a boost, andtherefore must be conjugate to a rotation γ ∈ SO(d). For d = 2, the only involutive (non-unit) such γ is the orthogonal symmetry with respect to the line directed by e0

(associated

with (z 7→ −1/z) ∈ PSL(2)). The statement is now trivial.

The parabolic isometries ϕ1, ϕ2 remaining as above, we consider now another case.

Proposition IV.4.1.3 The following three conditions are equivalent :

(i) the isometry ϕ1 ϕ2 is a boost (with fixed ideal points p′1, p′2) ;

(ii) the isometry ϕ2 ϕ1 is a boost (with fixed ideal points p′′1, p′′2) ;

(iii) there exist four pairwise distinct ideal points p′1, p′2, p′′1, p′′2 ∈ ∂H2 \ p1, p2 such that

p′j = ϕ1(p′′j ), ϕ2(p′j) = p′′j .

If these conditions hold, the quadrangles P ′0 := p1, p′1, p2, p

′′2 and P ′′0 := p1, p

′2, p2, p

′′1

are harmonic. If moreover the geodesics [p′1, p′2] and [p′′1, p

′′2] are disjoint, then

- ϕ1, ϕ2 are conjugate by means of some involutive isometry σ : ϕ1 = σ ϕ2 σ and σ2 = 1;

- there exist unique boundary points q1, q2 ∈ ∂H2 \ p1, p2 such that the quadrangles

Q′0 := p1, q1, p2, ϕ1(q1) and Q′′0 := p2, q2, p1, ϕ2(q2) be harmonic ;

- the geodesics [p1, q1] and [p1, ϕ1(q1)] are disjoint from the geodesics [p2, q2] and [p2, ϕ2(q2)];- the ideal points p1, p

′1, ϕ1(q1), q2, p

′2, p2, p

′′1, ϕ2(q2), q1, p

′′2 are all distinct and arranged in that

order along the boundary ∂H2 (seen as a circle). See Figure IV.2.

Proof Again it is sufficient, as for Proposition IV.4.1.1, to consider the half-plane R×R∗+ ,

with p1 ≡ ∞ and p2 ≡ 0 , ϕ1(z) = z + 2 , ϕ2(z) = zcz+1

, ϕ1 ϕ2(z) = (2c+1)z+2cz+1 , and

ϕ2 ϕ1(z) = z+2cz+2c+1

. Therefore ϕ1 ϕ2 is a boost if and only if c < −2 or c > 0 , and similarlyfor ϕ2 ϕ1 .


Figure IV.2: the Dirichlet polyhedron P0 = P(Γ2, q) in the case of Proposition IV.4.1.3

In this case, the fixed points are p′1, p′2 = 1 ±

√1 + 2

c and p′′1, p′′2 = −1 ±

√1 + 2

c . Clearly

p′j = ϕ1(p′′j ), whence ϕ2(p′j) = p′′j . Moreover P ′0 =∞, 1 +

√1 + 2

c , 0,−1 −√

1 + 2c

and

P ′′0 =∞, 1−

√1 + 2

c , 0,−1 +√

1 + 2c

are harmonic.

Reciprocally, if (iii) holds, then clearly ϕ1 ϕ2 fixes p′1, p′2 and then is conjugate to some

dilatation (z 7→ a2z), so that it can fix a point in H2 if and only if it is the identical map. Asϕ1, ϕ2 were supposed not to have the same fixed point, this cannot happen. Hence ϕ1 ϕ2 isindeed a boost.

Furthermore, the geodesics [p′1, p′2] and [p′′1, p

′′2] are disjoint if and only if c < −2 . Suppose

that this is the case. The claim about σ follows from the identity :(0 −

√2/|c|√

|c|/2 0

)×(

1 0c 1

)×(

0 −√

2/|c|√|c|/2 0

)−1

=

(1 2 sign(−c)0 1

).

Then Q′0 = ∞, q1, 0, q1 +2 and Q′′0 =

0, q2,∞, q2cq2+1

are harmonic if and only if q1 = −1

and q2 + q2cq2+1 = 0⇔ q2 = −2

c. As c < −2 , the last claims follow directly.


Remark IV.4.1.4 If the parabolic isometries ϕ1, ϕ2 satisfy ϕ1 = σ ϕ2 σ for some involu-tive isometry σ , then the isometry ϕ1 ϕ2 can be parabolic, or a boost with disjoint geodesics[p′1, p

′2] and [p′′1, p

′′2] (with the notations of Proposition IV.4.1.3), or a rotation.

Proof Once again we can suppose that p1 =∞ and p2 = 0 , ϕ1(z) = z+2 , ϕ2(z) = zcz+1

,

and ϕ1 ϕ2(z) = (2c+1)z+2cz+1

. As in the proof of Remark IV.4.1.2, we can take

σ = ±(

a −ba2+1b −a

), so that we have :

(a −b

a2+1b −a

)(1 0c 1

)= ±

(1 20 1

)(a −b

a2+1b −a

),

which is equivalent to a = 0 and c = −2b−2. Hence c can be any negative real number.

Proposition IV.4.1.5 Consider either the case of Proposition IV.4.1.1, or the case ofProposition IV.4.1.3, with disjoint geodesics [p′1, p

′2] and [p′′1, p

′′2]. Then the group generated

by the parabolic isometries ϕ1, ϕ2, say Γ2 , is free (with rank 2) and Fuchsian.

Proof In the case of Proposition IV.4.1.1, we have considered the harmonic quadrangleP0 = p1, p, p2, p

′. In the case of Proposition IV.4.1.3, with disjoint geodesics [p′1, p′2] and

[p′′1, p′′2], let us instead consider the convex polygon P0 bounded by the four sides [p1, q1],

[p1, ϕ1(q1)], [p2, q2] and [p2, ϕ2(q2)] (with the notations of Proposition IV.4.1.4), which arepair-wise disjoint (disregarding the common vertices p1, p2). Note that this P0 is alsobounded (at infinity) by two intervals (or arcs) of ∂H2 (seen as a circle), ϕ1(q1) q2 andϕ2(q2) q1 , joining respectively q1 to ϕ2(q2) and q2 to ϕ1(q1) and not containing the previ-ous ideal vertices, and is in fact the convex hull of

P0 := q1, p1, ϕ1(q1), q2, p2, ϕ2(q2) ∪ ϕ1(q1) q2 ∪ ϕ2(q2) q1 .

Thus the “harmonic” former case (which is a particular case of Theorem IV.3.1 with n = 2)appears as a limit of the “non-harmonic” latter, when q1, ϕ2(q2) merge into p′, and q2, ϕ1(q1)merge into p . The proofs for both cases can be unified, by agreeing that in the harmoniccase, both q1, ϕ2(q2) denote p′ and both q2, ϕ1(q1) denote p . The remaining of this proof,written for the non-harmonic case, will be similar to the proof of Lemma IV.3.2.

Observe that any parabolic isometry preserves the clockwise ordering of points on ∂H2 (seenas a circle). Hence, the four isometries ϕ1, ϕ

−11 , ϕ2, ϕ

−12 map P0 into the closed half-plane of

H2 not containing P0 and delimited by [p1, ϕ1(q1)], [p1, q1], [p2, ϕ2(q2)], [p2, q2] respectively.Therefore, the five convex polygons P0, ϕ1(P0), ϕ−1

1 (P0), ϕ2(P0), ϕ−12 (P0) have pair-wise

disjoint interiors, and their union is the convex hull of P0∪ϕ1(P0)∪ϕ−11 (P0)∪ϕ2(P0)∪ϕ−1

2 (P0).

Then the same procedure applies to each such polygon ϕεjj (P0), by means of the parabolic

isometries ϕεjj ϕ

εkk ϕ

−εjj fixing ϕ

εjj (pk), for 1 ≤ j, k ≤ 2 and εj , εk = ±1 (but εk 6= −εj if

k = j, not to return to P0 ). We thus obtain the 17 isometric convex polygons P0 , ϕεjj (P0),

ϕεjj ϕ

εkk (P0), which have pair-wise disjoint interiors.


Going so on by induction, at the m-th step we obtain (2× 3m − 1) isometric polygons

ϕε1j1 ϕε2j2. . . ϕεkjk (P0) , 0 ≤ k ≤ m, 1 ≤ ji ≤ 2 , εk = ±1 ,

indexed by the words of length ≤ m on the alphabetϕ±1

1 , ϕ±12

, which have pair-wise

disjoint interiors.

As a consequence, observe that for z in the interior of P0 and for any non-trivial reducedword ϕε1j1 ϕ

ε2j2. . . ϕεkjk , the point ϕε1j1 ϕ

ε2j2. . . ϕεkjk (z) never belongs to P0 . This immediately

implies the statement.

As a simple consequence of the preceding proof, we also obtain the following.

Corollary IV.4.1.6 In the case of Proposition IV.4.1.1, all ideal vertices are parabolic (i.e.,fixed by some parabolic element of Γ2), and there are exactly three classes of ideal verticesmodulo Γ2 (a set of representatives being p1, p2, p). Hence, the orbit space Γ2\H2 has threecusps (a cusp being the image of a parabolic vertex in the orbit space).

In the case of Proposition IV.4.1.3, there are exactly two classes of parabolic verticesmodulo Γ2 (a set of representatives being p1, p2), and the points of Γ2 qj (in the notationof Proposition IV.4.1.4) are not parabolic. Hence, the orbit space Γ2\H2 has two cusps.

Proof Only the last assertion deserves a complementary explanation. Clearly the points ofa same orbit are either all parabolic or all not. Recall from Proposition IV.4.1.3 that qj isfixed by a boost of Γ2 . Now, no point q can be fixed by both a boost ϕ and a parabolicisometry ψ of the same discrete group. Indeed, taking the half-plane R × R∗+ , up to aconjugation, we can suppose that q = ∞ , ϕ(z) = az and ψ(z) = z + u , with u ∈ R∗ anda > 1 or 0 < a < 1 . But then clearly ϕkψ ϕ−k | k ∈ Z does not constitute a discretesubset, yielding the wanted contradiction.

Remark IV.4.1.7 The Fuchsian group Γ2 contains boosts, but no rotation.

Proof By Lemma IV.2.1, a free Kleinian group cannot contain any rotation (which shouldbe a torsion element). The existence of boosts in Γ2 is obvious in the case of PropositionIV.4.1.3. In the case of Proposition IV.4.1.1, Remark II.2.3.5 allows us to consider the half-plane R×R∗+ , P0 = ∞,−1, 0, 1, ϕ1(z) = z + 2 , and ϕ2(z) = z

1−2z. Then ϕ−1

1 ϕ2 =(z 7→

5z−21−2z

)is a boost. This is again a particular case of Remark IV.3.6.

Theorem IV.4.1.8 Consider either the case of Proposition IV.4.1.1, or the case of Proposi-tion IV.4.1.3 with disjoint geodesics [p′1, p

′2] and [p′′1, p

′′2]. Then the convex polygon P0 (defined

in the proof of Proposition IV.4.1.5) happens to be the Dirichlet polyhedron P(Γ2, q), for anypoint q ∈ H2 of the geodesic joining the fixed points p1, p2 of the two parabolic isometriesϕ1, ϕ2 generating freely Γ2 .


Proof As for Proposition IV.4.1.5, let us unify the proof for both cases, by including thelimiting former case into the latter. By Remark II.2.3.5 and Propositions IV.4.1.1, IV.4.1.3, itis enough to consider the half-plane R×R∗+ , ϕ1(z) = z+2 , ϕ2(z) = z

cz+1 with c ≤ −2 , and

P0 bounded by the geodesics [∞,−1],[

2c , 0],[0,−2

c

], [1,∞] and by the intervals

[− 1, 2

c

],[

− 2c , 1]

(which vanish in the former case) of the horizontal boundary line R .

Consider then λ > 0 and the Dirichlet polygon P := P(Γ2, λ

√−1

), which by definition is

included in P ′ := P ′∞ ∩ P ′0 , with

P ′∞ :=z ∈ R×R∗+

∣∣∣dist(λ√−1 , z) ≤ dist

(ϕ±1

1 (λ√−1 ), z

)and

P ′0 :=z ∈ R × R∗+

∣∣∣ dist(λ√−1 , z) ≤ dist

(ϕ±1

2 (λ√−1 ), z

). Now, it is immediate that

P ′∞ =z ∈ R × R∗+

∣∣ |<(z)| ≤ 1

. And recalling the formula (II.1) for the hyperbolicdistance :

ch[dist

(x+ y

√−1 , n+ ν

√−1

)]=

(x− n)2 + y2 + ν2

2ν y,

since ϕε2(λ√−1 ) = εcλ2+λ

√−1

1+c2λ2 , we obtain(for ε = ±1 and z = x+ y

√−1

):

dist(λ√−1 , z) ≤ dist

(ϕε2(λ

√−1 ), z

)⇐⇒ x2+y2+λ2

1+c2λ2 ≤(x− εcλ2

1+c2λ2

)2+ y2 + λ2

(1+c2λ2)2

⇐⇒ (x− εc )

2 + y2 ≥ c−2 .

Hence we obtain P ⊂ P ′∞ ∩ P ′0 = P0 .

Conversely, for any q in the interior of P0 , by Proposition IV.2.2 there exists γ ∈ Γ2 suchthat γ(q) ∈ P, which implies in turn, by the above, that γ(q) ∈ P0 . Now, by the proof ofProposition IV.4.1.5, this forces γ to be the unit map, hence q ∈ P, concluding the proofthat P0 = P is a Dirichlet polygon for Γ2 .

Remark IV.4.1.9 Theorem IV.4.1.8 allows to complete somewhat the description of Corol-lary IV.4.1.6, of the orbit space in the the case of Proposition IV.4.1.3 : Γ2\H2 has two cusps,and a so-called funnel, which is an unbounded part of this Riemann surface, parametrizedby S1 × R+, with the perimeter of S1 × y equivalent to Cey as y → ∞. The circle atinfinity Γ2\

([q1, ϕ2(q2)]∪ [q2, ϕ1(q1)]

)bounds the funnel. Notice that the surface Γ2\H2 has

an infinite volume due to its funnel, contrary to what happens in the case of PropositionIV.4.1.1 (considered again in the following theorem). See Figure IV.2.

Theorem IV.4.1.10 In the case of Proposition IV.4.1.1, fixing the half-plane R×R∗+ andthe harmonic quadrangle ∞,−1, 0, 1, the Fuchsian group Γ2 of Proposition IV.4.1.5 is theprincipal congruence group Γ(2). Hence, Γ(2) is free, Fuchsian, has generator

z 7→ z + 2 , z 7→ z1−2z , and admits the Dirichlet harmonic convex quadrangle P0 , convex

hull of ∞,−1, 0, 1, as a fundamental polygon. See Figure IV.3.


∞

−1 0 1

Figure IV.3: harmonic quadrangle P0 , fundamental domain for Γ(2)

Proof By Lemma IV.4.1.1, we know that in the case of the statement Γ2 is generated byz 7→ z+2 and z 7→ z

1−2z, which belong to Γ(2). It remains to show that Γ(2) ⊂ Γ2 . Suppose

that we have some γ0 ∈ Γ(2) \ Γ2 . Then by Theorem IV.4.1.8 and Proposition IV.2.2, we

would have some γ1 ∈ Γ2 such that z0 := γ1γ0(√−1 ) ∈ P0 . Hence z0 = (2a+1)

√−1 +2b

2c√−1 +2d+1

, for

non all vanishing integers a, b, c, d such that 2(ad − bc) + a + d = 0 , by definition of Γ(2).

Thus, z0 =4(ac+ bd) + 2(b+ c) +

√−1

4(c2 + d2 + d) + 1=:

2m+√−1

4q + 1must belong to P0 . This means

that|2m| ≤ 4q + 1 and 1

4 ≤ |z0 ± 12 |2 ,

or equivalently|2m| ≤ 4q and 4m2 ± 2m(4q + 1) ≥ −1 ,

hence|m| ≤ 2q and [2m± (4q + 1)]m ≥ 0 ,

whence m = 0 . So far, we would have γ := γ1γ0 = ±(A BC D

)∈ Γ(2) \ Γ2 such that

AD −BC = 1 and AC +BD = 0 . Whence BD2 = −(1 +BC)C and then

C = −(C2 +D2)B , and 0 = [D− (C2 +D2)A]B . Now, B = 0 would imply AC = 0 andAD = 1 , whence C = 0 and γ = ±1 , which is excluded. Hence D = (C2 + D2)A , and


then (C2 + D2) divides 1, that is, (C2 + D2) = 1 , D = A , C = −B ; whence γ = ±1 or

γ = ±(

0 −11 0

)/∈ Γ(2) , which is excluded. This concludes the proof.

It is clear from Theorem IV.4.1.10 that taking n = 2 and the quadrangle P0 harmonicin Theorem IV.3.1, yields (a conjugate of) the principal modular group Γ(2).

IV.4.2 From Γ(2) to Γ(1)

Fix a harmonic quadrangle A,B,C,D (hence a fundamental domain for Γ(2), by The-orem IV.4.1.10) in H2, and denote its centre by Ω, and by σ ≡ [A ↔ C,B ↔ D] thesymmetry with respect to Ω. Let us consider also the parabolic isometry

τ ≡ [A 7→ A,B 7→ C,C 7→ D] , and :

A′ the orthogonal projection of A on CD , A′′ the orthogonal projection of A on BC ,

C ′ the orthogonal projection of C on AD , C ′′ the orthogonal projection of C on AB ,

β := AA′′ ∩ CC ′′, δ := AA′ ∩ CC ′. See Figure IV.4.

The following lemma is almost obvious, by considering the actions of σ, τ on A,B,C,D,and knowing that any isometry maps two orthogonal geodesics to two orthogonal geodesics.

Lemma IV.4.2.1 We have : β, δ ∈ BD, τ(β) = δ , τ(A′′) = A′, τ(Ω) = C ′, τ−1(Ω) = C ′′,σ(C ′′) = A′, σ(A′′) = C ′.

Consider now the following partition (up to the geodesic boundaries) of the fundamentaldomain delimited by A,B,C,D, into six sub-domains delimited by the quadrangles :

A,Ω, β, C ′′ ; A,Ω, δ, C ′ ; C,Ω, δ, A′ ; C,Ω, β, A′′ ; B,A′′, β, C ′′ ; D,C ′, δ, A′.By Lemma IV.4.2.1, we have τ(B,A′′, β, C ′′) = (C,A′, δ,Ω), τ(A,Ω, β, C ′′) = (A,C ′, δ,Ω),and under σ :

(A,Ω, β, C ′′)↔ (C,Ω, δ, A′) ; (A,Ω, δ, C ′)↔ (C,Ω, β, A′′) ; (B,A′′, β, C ′′)↔ (D,C ′, δ, A′) .

This proves the following.

Lemma IV.4.2.2 The 6 sub-domains listed above are isometric. (See Figure IV.4.)

Specialising now to the harmonic quadrangle A = ∞, B = −1, C = 0, D = 1 in the

half-plane R × R∗+ , we have (A,Ω, δ, C ′) =(∞,√−1 , e

√−1 π/3,

√−1 + 1

), and the convex

triangle (δ, β,A) becomes (see Figure IV.5) :

T :=z ∈ R×R∗+

∣∣∣ |<(z)| ≤ 12 , |z| ≥ 1

, delimited by

(e√−1 π/3, e

√−1 2π/3,∞

).


=

Figure IV.4: the 6 isometric sub-domains of Lemma IV.4.2.2

Moreover, σ can be identified now with [z 7→ −1/z] ∈ Γ(1), and τ with [z 7→ z+ 1] ∈ Γ(1).

We want to prove that T is a fundamental polygon for Γ(1), as will be stated in TheoremIV.4.1.10 below. To this aim, let us set

v := τ σ =[z 7→ z−1

z

]and Γ1 := Γ(2)

⋃σΓ(2)

⋃vΓ(2)

⋃σvΓ(2)

⋃v2Γ(2)

⋃σv2Γ(2).

We then have : v2(z) = −1z−1 , σ2 = v3 = 1 , σv(z) = −z

z−1 , σv2(z) = z − 1 , vσ(z) = z+ 1 .

It is very easily verified that σΓ(2)σ = Γ(2) and vΓ(2)v−1 = Γ(2), and then, that Γ1

is a subgroup of Γ(1), in which Γ(2) is normal. Moreover, by Theorem IV.4.1.10, Γ(2) isgenerated by (σv2)−2, (σv)2, so that Γ1 is the subgroup of Γ(1) generated by σ, v.Note that the six cosets defining Γ1 above are distinct

(all elements displayed above clearly

do not belong to Γ(2)), so that the quotient group Γ1/Γ(2) has order 6. Moreover, as

σvΓ(2) = v2σΓ(2), Γ1/Γ(2) is isomorphic to the (non Abelian) dihedral group of order 6.

By Lemma IV.4.2.2, this shows that the (convex hull of the) quadrangle A,Ω, δ, C ′ ofFigure IV.4 is a fundamental domain for Γ1 . Now T is plainly deduced from (A,Ω, δ, C ′)by translating its right half by −1, i.e., by applying σ2v to its right half. Hence the imagesof T under Γ1 , as well as the images of (A,Ω, δ, C ′) under Γ1 , cover the whole hyperbolicplane R × R∗+ . A fortiori, the images of T under Γ(1) cover the whole hyperbolic plane

R × R∗+ . This and the following lemma show that the interioroT of T is a fundamental

domain for the full modular group Γ(1).


Lemma IV.4.2.3 We have γ( oT)∩

oT = ∅ , for any γ ∈ Γ(1) \ 1.

Proof Consider z = <(z) +√−1 =(z) belonging to the interior

oT of T , and γ ≡

±(a bc d

)∈ Γ(1) \ 1 such that γ(z) ∈

oT . This implies in particular that γ cannot

be a translation (z 7→ z + b), hence that c 6= 0 . Then since |z| > 1 > 2∣∣<(z)

∣∣ we have :

|cz + d|2 > c2 − |cd|+ d2 = (|c| − |d|)2 + |cd| ≥ 1 , whence =(γ(z)

)==(z)

|cz + d|2 < =(z) .

Now the same holds with(γ(z), γ−1

), yielding also =(z) < =

(γ(z)

), a contradiction.

−1 0 1

T

Figure IV.5: Dirichlet triangle T for Γ(1)

Theorem IV.4.2.4 The convex triangle T is a fundamental polygon for the full modulargroup Γ(1), which is generated by σ, v, or alternatively by σ, τ

(recall that σ(z) = −1/z ,

v(z) = (z − 1)/z , τ(z) = z + 1), and admits the presentation

σ, v

∣∣σ2 = v3 = 1

.

The quotient group Γ(1)/Γ(2) is isomorphic to the (non Abelian) dihedral group of order 6.Moreover, the convex triangle T is the Dirichlet polygon P

(Γ(1), 2

√−1

).


Proof The above (beginning of this section IV.4.2) shows thatoT is a fundamental domain

for both Γ(1) and Γ1 ⊂ Γ(1), which must therefore be equal. Finally, the three sides of Tare immediately obtained as T ∩ τ(T ), T ∩ τ−1(T ), T ∩ σ(T ), thereby completing the proofthat T is a fundamental polygon for Γ(1).

As to the claim about the presentation, observe that any reduced word w written with σ, vcan be writen then w = σivjγ , with i ∈ 0, 1, j ∈ 0, 1, 2, and γ ∈ Γ(2). Note also thatwe have w = 1 if and only if w = γ = 1 , hence, by Theorem IV.4.1.10, if and only if theword w is trivial. This shows that Γ(1) admits the presentation

σ, v

∣∣σ2 = v3 = 1

.

Finally let us prove the last sentence of the statement. Denote by T the subgroup (isomorphicto Z) of Γ(1) generated by the translation τ = (z 7→ z + 1), and by σ the subgroup(isomorphic to Z/2Z) of Γ(1) generated by σ . By definition of Dirichlet polygons, we haveat once :

P(Γ(1), 2

√−1

)⊂ P

(T , 2√−1

)∩ P

(σ, 2√−1

).

Recalling the formula (II.1) for the hyperbolic distance :

ch[dist

(x+√−1 y , n+ ν

√−1

)]=[(x− n)2 + y2 + ν2

]/(2νy) ,

we immediately obtain P(T , 2√−1

)=|<(z)| ≤ 1

2

and P

(σ, 2√−1

)=|z| ≥ 1

, whence

P(Γ(1), 2

√−1

)⊂ T . Now by Proposition IV.2.2 P

(Γ(1), 2

√−1

)is a fundamental polygon

of Γ(1), as is T by Theorem IV.4.2.4. They must therefore be equal.

Note that the vertices e√−1 π/3, e

√−1 2π/3 of T are identified (in the orbit space Γ(1)\R×

R∗+ ) by the translation τ = vσ ∈ Γ(1), and are called elliptic points : they have non-trivialfinite stabilisers (in Γ(1)), 1, v, v2 and 1, σvσ, σv2σ respectively.

√−1 is another elliptic

point, having stabiliser 1, σ. The ideal vertex ∞ is parabolic, having as stabiliser theinfinite cyclic subgroup generated by the horizontal translation τ . Any other point of T hasa trivial stabiliser in Γ(1).

Recall from Corollary IV.4.1.6 that the image of the parabolic vertex ∞ in the orbit spaceΓ(1)\R×R∗+ is called a cusp. This name expresses the visual idea of a boundary point atinfinity, in the vicinity of which Γ(1)\R×R∗+ becomes infinitely thin. Thus, Γ(1)\R×R∗+is a simply connected non-compact surface having a unique cusp and finite volume, equal toπ/3 , which is the area of the triangle T

(calculated with respect to the hyperbolic volume

measure y−2dxdy of R×R∗+, recall Proposition III.6.2.2).

Both elliptic points e√−1 π/3 and

√−1 of T correspond to two singularities of the surface

Γ(1)\R × R∗+ , which is not a Riemann surface, but an orbifold. While Γ(2)\R × R∗+ isa Riemann surface, having 3 cusps and surface 2π , as is clear from Corollary IV.4.1.6 andTheorem IV.4.1.10.


IV.4.3 Plane tesselation by means of two boosts, and DΓ(1)

We consider now two boosts ψ1, ψ2 of the hyperbolic plane H2, with four distinct fixedpoints, p1, p

′1 and p2, p

′2 respectively, pj denoting the unstable (i.e., repulsive) fixed point

and p′j denoting the stable (i.e., attractive) fixed point.

We shall obtain an interesting example of Fuchsian group which does pertain neither to thoseexhibited by Theorem IV.3.1, nor to those obtained in Section IV.4.1.

Lemma IV.4.3.1 Suppose that the commutator [ψ1, ψ2] := ψ1 ψ2 ψ−11 ψ−1

2 is a parabolicisometry, with fixed ideal point q , and consider the ideal quadrangle

P ′0 :=q, ψ−1

2 (q), ψ−11 ψ−1

2 (q), ψ−11 (q)

.

Then P ′0 is harmonic if and only if ψ1, ψ2 are conjugate by means of some parabolic isometry.If this holds, then there exists an involutive isometry σ such that ψ−1

j = σ ψj σ and σ(pj) =

p′j for j = 1, 2 , and both quadranglesψ−1

1 (q), p1, ψ−12 (q), p′2

and

ψ−1

1 (q), p2, ψ−12 (q), p′1

are harmonic.

Proof Suppose that P ′0 is a harmonic quadrangle (and in particular that its four verticesare pairwise distinct). Take the Poincare half-plane model (recall Section II.3, page 51)R × R∗+ , with ψ−1

1 (q) ≡ 0 , ψ−12 (q) ≡ ∞ and q ≡ 1 . Hence ψ1(0) ≡ 1 ≡ ψ2(∞). Then

the assumption (P ′0 harmonic and [ψ1, ψ2](q) = q) forces −1 ≡ ψ−11 ψ−1

2 (q) = ψ−12 ψ−1

1 (q),and then ψ1(−1) ≡ ∞ , ψ2(−1) ≡ 0 . The isometry of Proposition II.2.3.1(v) exchangingthe vertices of P ′0 is the involution σ = [z 7→ −1/z]. The above constraints on ψ1, ψ2 forces

ψ1(z) =2z + 1

z + 1, ψ2(z) =

z + 1

z + 2, whence p1 = 1−

√5

2, p′1 = 1+

√5

2, p2 = −1−

√5

2, p′2 = −1+

√5

2.

Then the parabolic isometry h := [z 7→ z − 1] satisfies ψ2 = hψ1 h−1, and we have ψ−1

j =σ ψj σ . The quadrangles 0, p1 , ∞ , p′2 and 0, p2 , ∞ , p′1 are clearly harmonic.

Moreover we have σ(pj) = p′j .

Conversely, suppose that there exists a parabolic isometry h satisfying ψ2 h = hψ1 . Take thePoincare half-plane model R×R∗+ such that the fixed point of h is ∞ , so that h(z) = z− cfor some c ∈ R∗, and p1 = 1−

√5

2, p′1 = 1+

√5

2. Note that this is clearly possible, since h can

fix neither p1 nor p′1 (this would imply p1, p′1 ∩ p2, p

′2 6= ∅).

Then we have ψ1(z) =(1 + α)z + 1

z + α, with α ∈ 1,−2. Now α = −2 is excluded, since p1

would be attractive, for example since− 1+

√−5

2+1

1+√−5

2−2

= −4+√−5

7. Hence ψ1(z) =

2z + 1

z + 1. Then

ψ2(z) = ψ1(z + c)− c =(2− c)z + c− c2 + 1

z + c+ 1and


[ψ1, ψ2](z) =(2c3 + c2 − 11c+ 1)z + 2c4 − 3c3 − 11c2 + 18c

(c3 − 7c)z + c4 − 2c3 − 6c2 + 11c+ 1. Since [ψ1, ψ2] is parabolic, with

fixed point say q , we have

±2 = (2c3 +c2−11c+1)+(c4−2c3−6c2 +11c+1) = c4−5c2 +2 , i.e., c ∈

0,±√

5 ,±2,±1

,

and q = q(c) ≡ (2c3+c2−11c+1)−(c4−2c3−6c2+11c+1)2(c3−7c)

= c3−4c2−7c+222(7−c2)

= 2c2−11c2−7

− c2

. Now c 6= 0 ,

and (for ε = ±1) q(ε√

5 ) = 1−ε√

52 ∈ p1, p

′1, q(2ε) = 1− ε , q(ε) = 3

2 − ε2

.

The first case is excluded, since then the pair of fixed points of ψ2 would be

1−2ε√

5±√

52

,

which intersects p1, p′1. In the second case (c = 2ε), we obtain ψ2(z) = 2(1−ε)z+2ε−3

z+2ε+1 and

then P ′0 ≡

1− ε , 2−ε1−ε , − ε , −ε

ε+1

which is indeed harmonic. In the third case (c = ε), we

obtain ψ2(z) = 2(1−ε)z+2ε−3z+2ε+1 and then P ′0 ≡

3−ε

2, 2

1−ε , 1+ε−2

, 1−ε1+ε

which is indeed harmonic.

Now we have the analogue of Proposition IV.4.1.5.

Proposition IV.4.3.2 Consider, as in Lemma IV.4.3.1, two boosts ψ1, ψ2 having a para-bolic commutator [ψ1, ψ2]. Then the group Γ′2 generated by ψ1, ψ2 is free (with rank 2)and Fuchsian.

Proof Consider, as in Lemma IV.4.3.1, the fixed point q of [ψ1, ψ2], and the ideal quad-rangle

P ′0 :=q, ψ−1

2 (q), ψ−11 ψ−1

2 (q), ψ2 ψ−11 ψ−1

2 (q) = ψ−11 (q)

,

which we can see either as a quadruple of ideal vertices or as the convex polygon havingthese vertices. Observe now that any boost preserves the clockwise ordering of points on∂H2 (seen as a circle), so that the four isometries ψ1 , ψ

−11 , ψ2 , ψ

−12 map P ′0 into the closed

half-plane of H2 not containing P0 and delimited respectively by :[q, ψ−1

2 (q)],[ψ−1

1 (q), ψ−11 ψ−1

2 (q)],[q, ψ−1

1 (q)],[ψ−1

2 (q), ψ−11 ψ−1

2 (q)].

Hence, the five convex polygons P ′0, ψ1(P ′0), ψ−11 (P ′0), ψ2(P ′0), ψ−1

2 (P ′0) have pair-wisedisjoint interiors, and as union the convex hull of P ′0 ∪ψ1(P ′0)∪ψ−1

1 (P ′0)∪ψ2(P ′0)∪ψ−12 (P ′0).

Then (as in the proof of Proposition IV.4.1.5) the same procedure applies to each polygon

ψεjj (P ′0), by means of the boosts ψ

εjj ψεkk ψ

−εjj having parabolic commutators[

ψεjj ψ1 ψ

−εjj , ψ

εjj ψ2 ψ

−εjj

]= ψ

εjj [ψ1, ψ2]ψ

−εjj , for 1 ≤ j, k ≤ 2 and εj , εk = ±1 (but εk 6=

−εj if k = j, not to return to P ′0 ). We thus obtain the 17 isometric convex polygons P ′0 ,ψεjj (P ′0), ψ

εjj ψεkk (P ′0), which have pair-wise disjoint interiors. Continuing in this way by

induction, at the m-th step we obtain (2× 3m − 1) isometric polygons

ψε1j1 ψε2j2. . . ψεkjk (P ′0) , 0 ≤ k ≤ m, 1 ≤ ji ≤ 2 , εk = ±1 ,


indexed by the words of length ≤ m on the alphabet ψ±11 , ψ±1

2 , which have pair-wisedisjoint interiors.

As a consequence, observe that for z in the interior of P0 and for any non-trivial reducedword ψε1j1 ψ

ε2j2. . . ψεkjk , the point ψε1j1 ψ

ε2j2. . . ψεkjk (z) never belongs to P ′0 . This immediately

implies the statement.

Note that, since Γ′2 acts transitively on P ′0 , all ideal vertices of Γ′2 are equivalent andparabolic (recall from Corollary IV.4.1.6 that the so-called parabolic points of a Kleinian groupΓ are the vertices fixed by some parabolic element of Γ), so that the orbit space Γ′2\H2 hasa unique cusp.

The following proposition IV.4.3.3 has to be compared with Theorem IV.4.1.10 : twodifferent Kleinian groups

(which are here the two rank 2 free Fuchsian subgroups Γ(2) and

DΓ(1) of PSL(2,Z))

can admit the same fundamental polyhedron (which is here a harmonic

quadrangle), though they give rise to very different, non-isometric orbit spaces : Γ(2)\H2 hasthree cusps (and genus 0), while DΓ(1)\H2 has a unique cusp (and genus 1).

Actually, the two parabolic elements ϕ1 = [z 7→ z + 2], ϕ2 =[z 7→ z

1−2z

](of Proposi-

tion IV.4.1.10) generating Γ(2) on the one hand, and the two boosts ψ1, ψ2 (of PropositionIV.4.3.3) generating DΓ(1) on the other hand, realize two different pairings of the sides ofthe fundamental harmonic quadrangle P0 . See Figure IV.6. See also [Da].

Furthermore, Γ(2) and DΓ(1) differ algebraically by the facts that Γ(1)/DΓ(1) is cyclicwhile Γ(1)/Γ(2) is not Abelian.

Proposition IV.4.3.3 Considering the half-plane R×R∗+ , the harmonic quadrangle P ′0 =

0,−1,∞, 1 = P0 , and ψ1 =

[z 7→ 2z + 1

z + 1

], ψ2 =

[z 7→ z + 1

z + 2

]∈ Γ(1), we obtain a real-

ization of the group Γ′2 of Proposition IV.4.3.2, which happens to be the group DΓ(1) =[Γ(1),Γ(1)], generated by the commutators of Γ(1) = PSL(2,Z). Moreover, Γ(1)/DΓ(1) is acyclic group of order 6.

Proof Recall from Theorem IV.4.2.4 that σ = [z 7→ −1/z] and τ = [z 7→ z + 1] = vσgenerate Γ(1) = PSL(2,Z). Then ψ1 = [τ, σ] = τ σ τ−1 σ−1 ∈ DΓ(1), and ψ2 = [σ, τ−1] ∈DΓ(1). Hence Γ′2 ⊂ DΓ(1) . On the other hand, since Γ(1)/DΓ(1) is the Abelianized groupof Γ(1), we know from Theorem IV.4.2.4 that it admits the presentation

σ, v | σ2 = v3 =

1 = [σ, v]

, hence that it is cyclic of order 6, generated by vσ = τ . Hence, we can obtaina fundamental polygon for DΓ(1) by juxtaposing 6 isometric images of T , for example⋃0≤k≤5

τkT . Finally, we find the index [DΓ(1) : Γ′2] = [Γ(1) : Γ′2]/[Γ(1) : DΓ1] = 6/6 = 1 , so

that DΓ(1) = Γ′2 .


∞

Figure IV.6: fundamental domain P0 for Γ(2) and DΓ(1), with two distinct pairings

DΓ(1) is not a principal congruence group, but it is however a congruence group, i.e., a(modular) group containing a principal congruence group. Indeed we have Γ(6) ⊂ DΓ(1),with DΓ(1)/Γ(6) isomorphic to Z/6Z× Z/2Z . More precisely, we have

DΓ(1) =⋃

−3≤k≤2

[ψk1Γ(6)

⋃ψk+1

1 ψ2Γ(6)]

(see [F3]).

Note that the subgroup [DΓ(1), DΓ(1)] is not modular, having infinite index in Γ(1), sinceDΓ(1)/[DΓ(1), DΓ(1)] is isomorphic to Z2. Hence, the normal subgroup [Γ,Γ] of a modulargroup Γ can be a free group, but is generally not modular.

The proof of Proposition IV.4.3.3 exhibits another fundamental polygon for DΓ(1), namelythe fundamental octogon D :=

⋃0≤k≤5

τkT of Figure IV.7.

IV.4.4 Plane tesselation yielding Γ(3)

Here is another application of Theorem IV.3.1, which we give without its (somewhatlengthy) proof, left as an exercise.

Theorem IV.4.4.1 For n = 3 and the hexagon P0 =∞,−1,−2

3 , 0,23 , 1

(in the Poinca-re upper half-plane model R × R∗+ ), Theorem IV.3.1 yields the principal congruence group


Figure IV.7: fundamental octogon D for DΓ(1)

Γ(3), freely generated by the parabolic isometries :

ϕ−1 =

[z 7→ −2z − 3

3z + 4

], ϕ0 =

[z 7→ z

3z + 1

], ϕ1 =

[z 7→ 4z − 3

3z − 2

],

admitting the fundamental hexagon P0 . See Figure IV.8. The orbit space Γ(3)\R×R∗+ isa Riemann surface having 4 cusps, surface 4π , and genus 0. Moreover, the quotient groupΓ(1)/Γ(3) has order 12, and is isomorphic to the alternating group A4 (of even permuta-tions on 4 points, also isomorphic to the group of Euclidean rotations preserving a regulartetrahedron in R3).

Recall from Corollary IV.4.1.6 that a cusp is a class of equivalent (under the action ofthe considered Kleinian group) parabolic vertices. For example the orbit space Γn\H2 ofTheorem IV.3.1 has (n+ 1) cusps.

Remark IV.4.4.2 Theorem IV.3.1 cannot give rise to the the principal congruence groupΓ(n) when n ≥ 4 (though it is still a free group), as it did in Theorems IV.4.1.10 andIV.4.4.1. For example, Γ(4) has rank 5 (i.e., is freely generated by 5 elements), and 6inequivalent parabolic points, and Γ(5) has rank 11, and 12 inequivalent parabolic points.

More generally, for any n ≥ 3 , Γ(n) is free with rank n2

4 (n6 + 1)∏p|n

(1 − p−2), and has

n2

2

∏p|n

(1 − p−2) inequivalent parabolic points(and genus 1 + n2

4 (n6 − 1)∏p|n

(1 − p−2)). See(

[Le], Section XI.3D)

and([Mi], Section 4.2

).

In the case of a generic modular group Γ, the genus (which is the maximal number of non-intersecting closed curves which can be drawn on the surface Γ\H2 without disconnecting it)and the volume of Γ\H2 can be expressed in terms of three parameters, the numbers ν∞(Γ),


∞

−1 0 1

Figure IV.8: fundamental hexagon P0 for Γ(3)

ν2(Γ) and ν3(Γ), of Γ-inequivalent cusp points of Γ, of Γ-inequivalent elliptic points of Γ,of order 2 and 3, respectively. Of course, ν∞(Γ) is also the number of cusps of Γ\H2.

The genus of Γ\H2 is given by the formula([Mi], Theorem 4.2.11

):

g(Γ) = 1 + 112 [Γ(1) : Γ]− 1

4 ν2(Γ)− 13 ν3(Γ)− 1

2 ν∞(Γ) . (IV.2)

The volume of Γ\H2 is given by the formula([Mi], Theorem 2.4.3

):

V (Γ\H2) = 2π ×(2 g(Γ)− 2 + ν∞(Γ) + 1

2 ν2(Γ) + 23 ν3(Γ)

). (IV.3)

In the particular case of principal congruence groups Γ(N), very explicit formulae for[Γ(1) : Γ(N)] and for νj(Γ(N)) are known

(see ([Mi], Section 4.2), and Remark IV.4.4.2

).

IV.5 Notes and comments

See in ([Rac], Section 7.1) the example of a regular ideal dodecahedron, as funda-mental domain for the Kleinian group generated by the reflexions in its sides. Examplesin higher dimensions d ≥ 3 are often significantly more involved. Maybe the simplest

IV.5. NOTES AND COMMENTS 139

is PSO(1, d,Z) (made of the Lorentzian matrices with integer entries), which is cofi-nite with a number of cusps at least of order dd

2/4, and for which a nice algorithm toconstruct a fundamental domain is known by G. Collinet, using the lightlike vectorshaving integer coordinates and small Euclidean lengths.

In three dimensions, by means of Proposition I.1.5.2, it is equivalent to consider

sub-groups of PSL2(C). The action of γ := ±(a bc d

)∈ PSL2(C) on H3 ≡ C×R∗+ is

as follows :

γ(z, ζ) :=

(ζ

|cz − d|2 + ζ2|c|2,(d− cz)(az − b)− ζ2ca

|cz − d|2 + ζ2|c|2)

.

R. Swan [Sw] studied in a detailed way 3-dimensional analogues of modular groups,associated to quadratic imaginary number fields Q

[√−m ]. In particular the followingone.

Proposition IV.5.1 ([Sw]) The group PSL2

(Z +√−2Z

)is Kleinian, generated by

τ := ±(

1 10 1

), ν := ±

(1√−2

0 1

), α := ±

(0 −11 0

), with complete system of rela-

tions : τν = ντ , α2 = (τα)3 = (αν−1αν)2 = ±1 , and admits the fundamentalpolyhedron P :=

(z, ζ) ∈ C× R∗+∣∣∣ ∣∣<(z)

∣∣ ≤ 12

,∣∣=(z)

∣∣ ≤ 1√2

, min|z|, |z ± 1|, |z ±

√−2 |

2+ ζ2 ≥ 1

.

Note that this example (as that of the dodecahedron) is not cocompact, but cofinite,since P ⊂ ζ ≥ 1

2.

Remark IV.5.2 Cocompact examples of Fuchsian groups and domains (includinghow to obtain them by the Poincare polyhedron theorem) can be seen in ([Rac], Section11.2).

Examples of Fuchsian groups with fundamental domains having infinite volume (be-yond the above example of Proposition IV.4.1.3 and Remark IV.4.1.9), namely theso-called Schottky groups, can be found in [Da].

This chapter was included for self-containedness. It provides severalexamples in which the theory of the subsequent chapters V and VIII ap-ply, but has not the ambition to be a systematic introduction to Riemannsurfaces. Historical presentations are given in [Mr] and [SG].


Classical references such as [FK] include a lot of examples of funda-mental domains. See also [Kr] , [Rag], [Mas], [Rac], [MT], [Da]. Modulargroups are specifically studied in [Le], [Sc], [Mi]. Different notions of ge-ometrical finiteness are compared thoroughly in [B1], [B2].

Chapter V

Measures and flows on Γ\Fd

We consider here a fixed Kleinian group Γ and the associated relativemeasures and flows (induced by the measures of Section III.6 and theflows of Section II.3) on the orbit or quotient space Γ\Fd, and we inves-tigate their basic properties, when Γ is geometrically finite and cofinite.

We start with general notions about left quotients, Γ-invariant setsand covolume (Proposition V.1.1).

The first main result of this chapter is a mixing theorem for the ac-tion of the geodesic and horocyclic flows on square integrable Γ-invariantfunctions, namely Theorem V.3.1. It proceeds essentially from the com-mutation relations, that produce the instability of the flow illustrated byProposition II.3.9.

The second main result of this chapter is a Poincare inequality (i.e.,the existence of a spectral gap) for the Laplacian acting on Γ-invariantfunctions, namely Theorem V.4.2.5. We prove it by decomposing a fun-damental domain into a compact core and cusps neighbourhoods (calledsolid cusps) which overlap. Then we show that a Poincare inequalityholds in these parts, and we establish a general proposition stating thatit is conserved by taking union of overlapping domains.

141

142 CHAPTER V. MEASURES AND FLOWS ON Γ\Fd

V.1 Measures of Γ-invariant sets

Recall that any γ ∈ PSO(1, d) acts as well on the frame bundle Fd,and that plainly, if D ⊂ Hd is a fundamental domain for Γ, then thecylindrical domain π−1

0 (D) ⊂ Fd is fundamental for the (left) action ofΓ on Fd.

From now on, we shall consider mainly Γ-invariant functions. We firstspecify the relative (volume and Liouville) measures against which weshall integrate them. We also emphasize the invariance of these relativemeasures under the flows, which follows from the fact that the Kleiniangroup Γ acts on the left hand side, while the flows act on the right handside, so that these two actions commute.

Proposition V.1.1 Fix some Kleinian group Γ. Then the followingstatements hold.

(i) The volume of a fundamental domain D depends only on the Klei-nian group Γ, and will be called the covolume of Γ, and denoted bycovol(Γ).

(ii) The volume measure dp of Hd, restricted to a fundamental domainD, induces a measure dΓp on left Γ-invariant Borel sets of Hd, whichdepends only on Γ.

(iii) The Liouville measure λ of Fd, restricted to a cylindrical funda-mental domain π−1

0 (D), induces a relative Liouville measure λΓ on leftΓ-invariant Borel sets of Fd, which depends only on Γ.

(iv) The Liouville measure λΓ is invariant under the (right) action ofthe geodesic and horocycle flows, and more generally, under the rightaction of any g ∈ PSO(1, d).

Notation We denote henceforth by Γ\Fd the quotient of Fd underthe left action of the Kleinian group Γ, and similarly, by Γ\Hd the

V.1. MEASURES OF Γ-INVARIANT SETS 143

quotient of Hd. Then the Borel sets of Γ\Fd (respectively Γ\Hd) areidentified with the Γ-invariant Borel sets of Fd (respectively of Hd), andthe measure λΓ (respectively dΓp) is identified with a measure on Γ\Fd(respectively on Γ\Hd).

Proof Let D,D′ be fundamental domains for Γ. Observe that Hd \⋃g∈Γ

gD ⊂⋃g∈Γ

g ∂D ,

so that the volume of Hd \⋃g∈Γ

gD must be 0. Hence, for any non-negative measurable (left)

Γ-invariant function h on Hd we have :∫Dh(p)dp =

∫D∩

⋃g∈Γ gD

′h(p)dp =

∑g∈Γ

∫D∩ gD′

h(p)dp =∑g∈Γ

∫g−1D∩D′

h(p)dp =

∫D′h(p)dp .

Taking h ≡ 1 proves that the volumes of D and D′ are equal, proving (i). And this shows

more generally that

∫h(p) dΓp :=

∫Dh(p) dp is indeed uniquely defined, proving (ii).

(iii) Fix any (left) Γ-invariant test-function f on Fd, and two fundamental domainsD,D′ for Γ. Using the invariance of λ (recall Section III.6.2) and of f and proceeding asabove, we have :∫f × 1D π0 dλ =

∑g∈Γ

∫f 1D∩ gD′ π0 dλ =

∑g∈Γ

∫f 1g−1D∩D′ π0 dλ =

∫f × 1D′ π0 dλ .

(iv) Let us prove now the invariance under the flows, and more generally, that the rightaction of any g ∈ PSO(1, d) preserves the relative Liouville measure λΓ. Fix a Γ-invarianttest-function f on Fd, g ∈ PSO(1, d), and for any β ∈ Fd, set fg(β) := f(βg). Then forany cylindrical fundamental domain D := π−1

0 (D), Dg = π−10 (Dg) is another cylindrical

fundamental domain for the action of Γ on Fd, so that the right invariance of λ implies :∫f dλΓ =

∫Df dλ =

∫Dgfg dλ =

∫fg dλΓ.

A Kleinian group Γ is said cocompact if it has a relatively compactfundamental domain.

A Kleinian group Γ will be said cofinite if it has a finite covolume.


V.2 Ergodicity

Let us first introduce the important mixing and ergodic properties.

Definition V.2.1 Consider any probability space (E, E , µ) and a one-parameter group (gt) of measure-preserving maps such that for any f ∈L2(E , µ), t 7→ 〈f gt , f〉L2 is continuous.

The action of (gt) is said to be mixing if for any f, ϕ ∈ L2,

limt→±∞

〈f gt , ϕ〉L2 = 〈f, 1〉L2〈1, ϕ〉L2.

A function f ∈ L2 is said to be (gt)-invariant if f gt = f almost surelyfor any t ∈ R.

The action of (gt) is said to be ergodic if any (gt)-invariant f ∈ L2(µ)is almost surely constant.

We have the following very easy implication.

Proposition V.2.2 If (gt) is mixing, then it is ergodic.

Recall the fundamental ergodic theorem, due to Birkhoff. It statesthat if ergodicity holds, then temporal means converge to spatial mean.

Theorem V.2.3 (Ergodic Theorem) If (gt) is ergodic and if (t, β) 7→gt(β) is measurable on R× Γ\Fd, then for any f ∈ L2 we have :

limt→∞

1

t

∫ t

0

f gs ds =

∫f dµ , µ-almost surely and in L2.

Proof(F. Riesz, Yosida-Kakutani ; see also [Ka]

)Set F :=

∫ 1

0

f gs ds .

V.2. ERGODICITY 145

1) (Garsia) For any n ∈ N∗, set Fn := supk∈1,...,n

∫ k

0

f gs ds . As

∫ k+1

0

f gs ds =

∫ 1

0

f gs ds+

[ ∫ k

0

f gs ds] g1 ,

we have Fn ≤ Fn+1 ≤ F + F+n g1 , whence∫

Fn>0F dµ ≥

∫Fn>0

Fn dµ−∫F+n g1 dµ = 0 .

Since ‖F‖2 ≤ ‖f‖2 <∞ , letting n→∞ by dominated convergence weobtain the so-called “maximal ergodic lemma” :∫

supn≥1

1n

n−1∑k=0

Fgk >0

F dµ =

∫supk∈N∗

∫ k0fgs ds>0

F dµ ≥ 0 .

2) For any rational numbers a < b , consider the following (gt)-invariantmeasurable set

Eab :=

lim infn→∞

1

n

n−1∑k=0

F gk < a < b < lim supn→∞

1

n

n−1∑k=0

F gk.

Applying the above maximal ergodic lemma, with µ replaced by itsrestriction to Eab , successively to (F − b) and to (a− F ) yields :∫Eab

(F − b) dµ ≥ 0 and

∫Eab

(a− F ) dµ ≥ 0 , whence

µ(Eab) =−1

b− a

∫Eab

(F − b+ a− F ) dµ ≤ 0 , and then µ(Eab) = 0 .

This proves the µ-almost sure convergence of the sequence

1

n

∫ n

0

f gs ds =1

n

n−1∑k=0

F gk . Applying this convergence to f+ instead

of f and using(with [t] := maxn ∈ N |n ≤ t, for any t > 0

)

146 CHAPTER V. MEASURES AND FLOWS ON Γ\Fd∫ [t]

0

f+ gs ds ≤∫ t

0

f+ gs ds ≤∫ [t]+1

0

f+ gs ds , we deduce at once

the µ-almost sure existence of limt→∞

1

t

∫ t

0

f+ gs ds , and similarly of

limt→∞

1

t

∫ t

0

f− gs ds , hence of h := limt→∞

1

t

∫ t

0

f gs ds .

Clearly the measurable function h is (gt)-invariant, and belongs to L2

by Fatou’s Lemma, hence is µ-almost surely constant by the ergodicityassumption.

Finally for any µ-measurable set A and any t, R > 0 we have :∫A

∣∣∣∣1t∫ t

0

f gs ds∣∣∣∣2dµ ≤ 1

t

∫ t

0

∫A

|f gs|2 dµ ds

=1

t

∫ t

0

∫gs(A)

|f |2 dµ ds ≤ R2µ(A) +

∫|f |>R

|f |2 dµ

which(by taking a large enough R and then a small enough µ(A)

)shows

the uniform integrability of the family

(1

t

∫ t

0

f gs ds)2

t>0

.

The convergence to h in L2(µ) follows, and then

h =

∫h dµ = lim

t→∞

∫1

t

∫ t

0

f gs ds dµ = limt→∞

1

t

∫ t

0

∫f dµ ds =

∫f dµ .

V.3 A mixing theorem

Consider a cofinite Kleinian group Γ, the normalised induced measureµ := λΓ/|λΓ| = covol(Γ)−1 λΓ, the Hilbert space L2 ≡ L2(Γ\Fd, µ), with

inner product 〈f, ϕ〉L2 :=

∫f ϕ dµ , and a one-parameter subgroup (gt)

of PSO(1, d).

V.3. A MIXING THEOREM 147

Theorem V.3.1 The geodesic flow (θt) and any non-trivial one-para-meter subgroup (θ+

tu) of the horocycle flow are mixing, hence ergodic.

The proof of this theorem needs several steps. We mainly follow anelegant and elementary proof by Y. Guivarc’h [Gu], which shows in facta similar but stronger result : the above theorem is true not only forPSO(1, d), but even for SL(d + 1). Focussing on PSO(1, d) allows tosimplify partly the proof (when d > 2).

Consider the unitary representation γ 7→(f 7→ γf := f γ

), from

PSO(1, d) into the group of unitary endomorphisms of

L20 = L2

0(µ) =f ∈ L2

∣∣ 〈f, 1〉L2 = 0

.

Note that this representation has no non-zero PSO(1, d)-invariant vec-tor : indeed, since the action of PSO(1, d) is transitive, any PSO(1, d)-invariant function f ∈ L2

0 has to be constant, and then equal to zero.

We shall say that a sequence (γn = %nθrn%′n) in PSO(1, d) (using the

Cartan decomposition, recall Theorem I.7.1) goes to infinity if and onlyif rn →∞ . It is clear from the examples following Theorem I.7.1 thatthe one-parameter subgroups (θt) and (θ+

tu) (for any non-zero u ∈ Rd−1)go to infinity as t → ±∞ , so that the proof of the theorem followsimmediately from :

Proposition V.3.2 If (γn) ⊂ PSO(1, d) is a sequence going to infinity,then for any f, ϕ ∈ L2

0 we have : limn→∞〈γnf, ϕ〉L2 = 0 .

To establish Proposition V.3.2, we shall use the following lemma.

Lemma V.3.3 Suppose that there exists no non-zero vector f ∈ L20

fixed by all elements of Td−1 =θ+u ≡ (f 7→ f θ+

u )∣∣u ∈ Rd−1

.

Then for any f , ϕ ∈ L20 , we have : lim

t→∞〈θt f , ϕ〉L2 = 0 .


Proof For any f , ψ ∈ L20 and u ∈ Rd−1, using the commutation

relation (I.16) we have :∣∣∣⟨(θ+u ψ − ψ), θt f

⟩L2

∣∣∣ =∣∣∣⟨(θ+

u − 1)θtθ−t ψ, θt f⟩L2

∣∣∣=∣∣∣⟨θ−t ψ, θ−t(θ+

−u − 1)θt f⟩L2

∣∣∣ ≤ ‖θ−t ψ‖L2 ×∥∥θ−t(θ+

−u − 1)θt f∥∥L2

= ‖ψ‖L2 ×∥∥θ+−u e−t f − f

∥∥L2 −→ 0 .

Now the vector space V generated by

(θ+u ψ − ψ)

∣∣ψ ∈ L20 , u ∈ Rd−1

is dense in L2

0 . Indeed, if h ∈ V ⊥, then for any u ∈ Rd−1 we have :〈ψ, θ+

−u h〉L2 = 〈θ+u ψ, h〉L2 = 〈ψ, h〉L2 for any ψ ∈ L2

0 , whence θ+−u h = h .

By the hypothesis, this forces h = 0 . The statement follows at once.

Lemma V.3.4 Let (γn = %n θrn %n) ⊂ PSO(1, d) be a sequence suchthat lim

n→∞〈θrnf, ϕ〉L2 = 0 for any f, ϕ ∈ L2

0 . Then limn→∞〈γnf, ϕ〉L2 = 0 ,

for any f, ϕ ∈ L20 .

Proof Fix f, ϕ ∈ L20 . Up to extracting subsequences, we may suppose

that %n → % ∈ SO(d), %n → % ∈ SO(d), and 〈γnf, ϕ〉L2 → ` ∈ R .Then we obtain : |〈γnf, ϕ〉L2|

≤ |〈θrn %n f, (%−1n −%−1)ϕ〉L2|+|(%n−%)f, θ−rn %

−1ϕ〉L2|+|〈θrn % f, %−1ϕ〉L2|

≤ ‖f‖L2×‖L2(%−1n −%−1)ϕ‖L2 +‖(%n− %)f‖L2×‖ϕ‖L2 + |〈θrn % f, %−1ϕ〉L2|

which goes to 0. This shows that 0 is the only limit of the sequence〈γnf, ϕ〉L2 , whence the result.

Now, applying the above lemmas V.3.3 and V.3.4, we see that theproof reduces to proving that there exists no non-zero vector fixed by allelements of Td−1. As we already noticed that there exists no non-zeroPSO(1, d)-invariant vector, it reduces to the following.

V.3. A MIXING THEOREM 149

Lemma V.3.5 Any function f ∈ L20 such that θ+

u f ≡ f θ+u = f (for

any u ∈ Rd−1) is invariant under the right action of PSO(1, d).

Proof 1) Fix f ∈ L20 such that f θ+

u = f , for any u ∈ Rd−1, andsuch that ‖f‖L2 = 1 , and consider the continuous real valued functionΦ defined on G := PSO(1, d) by :

Φ(γ) := 〈f γ , f〉L2 .

We know that Φ(1) = 1, that Φ(γ) = Φ(γ−1), and that Φ(γ) =Φ(γ θ+

u ) = Φ(θ+u γ), for any real number u and any γ ∈ G ; and we

want to show that Φ ≡ 1 on G .

2) Let us begin with d = 2, and use the Iwasawa decomposition inPSO(1, 2) (recall Theorem I.7.2, applied to γ−1) : any element can bewritten as Rα θr θ

+σ (writing θ+

x , x ∈ R, for θ+x e2

) in a unique way. HereRα denotes the rotation by α in the oriented plane (e1,−e2). Fromthe hypothesis, we have for any reals α, r, x :

Φ(Rα θr θ+x ) = Φ(θ+

x Rα θr) = Φ(Rα θr) =: ϕ(α, r)

for some continuous function ϕ on (R/2πZ)×R , such that ϕ(0, 0) = 1 .

Let us first show that for any α, β /∈ πZ and t, r ∈ R , we have :

(1− cosα) er = (1− cos β) et =⇒ ϕ(α, r) = ϕ(β, t) .

Now it is enough to show that :

(1− cosα) er = (1− cos β) et =⇒ (∃x, x′ ∈ R) θ+x Rα = Rβ θ

+x′ θt−r ,

since by the commutation relation (I.16), we have :

θ+x Rα = Rβ θ

+x′ θt−r =⇒ θ+

x Rα θr = Rβ θ+x′ θt = Rβ θt θ

+e−tx′

=⇒ ϕ(α, r) = ϕ(β, t) .


Now for α, β /∈ πZ and s ∈ R such that (1 − cosα) es = (1 − cos β),the existence of real numbers x, x′ such that θ+

x Rα = Rβ θ+x′ θs follows

at once from Proposition II.6.3(i). This proves the desired formula.

3) Let us now deduce that necessarily Φ ≡ 1 on PSO(1, 2).

For any real r < t and ε > 0 , by continuity there exists α ∈ ]0, π[ suchthat β ∈ ]0, α] implies |ϕ(0, r)−ϕ(β, r)| < ε/2 and |ϕ(0, t)−ϕ(β, t)| <ε/2 . Therefore, choosing β such that (1 − cos β) = (1 − cosα) er−t <(1− cosα), we have :

|ϕ(0, r)− ϕ(0, t)| < ε+ |ϕ(α, r)− ϕ(β, t)| = ε ,

by 2) above. Hence for any real r, t we obtain ϕ(0, r) = ϕ(0, t), whenceϕ(0, r) = ϕ(0, 0) = 1.

By the definition of ϕ in 2) above, this means that Φ(θr) = 1, thereforethat f θr = f (for any r ∈ R), and then that

Φ(Rα θr θ+x ) = Φ(Rα θr) = 〈f Rα , f θ−r〉L2 = Φ(Rα) =: ψ(α) ,

for some even continuous function ψ on (R/2πZ) such that ψ(0) = 1 .

Since Φ(γ) = Φ(γ−1) for any γ ∈ G by definition of Φ , we haveΦ(θr θ

+x Rα) = ψ(α) as well. Now, for any α ∈ ]0, π[ , setting

rα := argsh (cotgα), we have log ch rα = − log sinα and then by Propo-sition II.6.3(ii) :

θ−rαRα = Rπ2θlog sinα θ

+−cotgα .

By the above, this entails that

ψ(α) = Φ(θ−rαRα) = Φ(Rπ

2θlog sinα θ

+−cotgα

)= ψ(π/2).

By continuity, this holds for α ∈ [0, π]. Therefore we obtain ψ(α) =ψ(0) = 1 for any α , that is : Φ ≡ 1 on PSO(1, 2), as claimed.

4) The above proves that any (θ+x e2

)-invariant function f ∈ L20 is also

(θt)-invariant and (Rα)-invariant. Now we can do exactly the same for

V.4. POINCARE INEQUALITY 151

any j ∈ 3, . . . , d, using θ+x ej|x ∈ R instead of θ+

x e2|x ∈ R, and

considering the rotations Rjα in the plane (e1, ej) instead of the rotations

Rα : the elements Rjα θr θ

+xej

constitute clearly a group isomorphic toPSO(1, 2), under which f is invariant.

Thus we see that any Td−1-invariant function f ∈ L20 is also (θt)-

invariant and invariant under any rotation Rjα . Now, observe that these

rotations Rjα (for 2 ≤ j ≤ d and α ∈ R) generate SO(d) : indeed, the

Lie algebra of the subgroup they generate contains all matrices E1,j (re-call Section I.3), and then also [E1,j, E1,k] = Ek,j , hence the whole so(d).Finally, any Td−1-invariant function f ∈ L2

0 is indeed also G-invariant,due to the Iwasawa decomposition (Theorem I.7.2).

V.4 Poincare inequality

We shall need only the case of hyperbolic domains, i.e., of connectedopen subsets of Hd, endowed with the restriction of the volume measuredp (recall Definition III.6.2.1). However it happens to be easier to beginwith the Euclidean case, as long as the Lebesgue measure is providedwith weights. Therefore we shall consider first the Euclidean framework.

V.4.1 The Euclidean case

We are interested in the following basic type of functional inequality.

Definition V.4.1.1 Consider a domain D ⊂ Rd and two positive mea-surable functions on D : ϕ bounded and ψ bounded away from 0. APoincare inequality I(D,ϕ, ψ) holds when there exists C = C(D,ϕ, ψ)such that for any integrable function f of class C1 on D satisfying

152 CHAPTER V. MEASURES AND FLOWS ON Γ\Fd∫D

f(x)ϕ(x) dx = 0 , we have :∫D

f(x)2 ϕ(x) dx ≤ C

∫D

|df |2(x)ψ(x) dx , (V.1)

where |df | denotes the Euclidean norm of the differential of f (i.e., ofits Euclidean gradient) and dx the Lebesgue measure of Rd.

Note that provided ϕ ∈ L1(D), I(D,ϕ, ψ) is clearly equivalent to theexistence of C such that for any f ∈ L1(D) ∩ C1(D),∫

D

f(x)2ϕ(x)dx ≤ C

∫D

|df |2(x)ψ(x) dx+C

(∫D

f(x)ϕ(x)dx

)2

. (V.2)

The following statement has the following interpretation : a spectral gap I(D,ϕ, ψ) yieldsa positive bottom for the spectrum of the induced theory obtained by a so-called killingprocedure on a non-empty open set U ⊂ D

(provided ϕ ∈ L1(D)

).

Proposition V.4.1.2 Consider any Borelian subset U of a domain D ⊂ Rd on which aPoincare inequality I(D,ϕ, ψ) holds. Then for any function h ∈ L1

(D,ϕ(x) dx

)∩ C1(D)

which vanishes on U , we have :∫Dh2(x)ϕ(x) dx ≤

∫D ϕ(x) dx∫U ϕ(x) dx

× C(D,ϕ, ψ)

∫D|dh|2(x)ψ(x) dx . (V.3)

Proof Suppose that ϕ ∈ L1(D), denote by m the normalized measure 1D(x)ϕ(x) dx∫D ϕ(x) dx

, and

set V := D \ U and f := h−∫Dh dm . Applying (V.1) yields :

C∫D ϕ

∫D|dh|2(x)ψ(x) dx = C∫

D ϕ

∫D|df |2(x)ψ(x) dx ≥

∫f2 dm =

∫h2 dm −

[∫h dm

]2

=

∫Vh2 dm−

[m(V )

∫Vh dmm(V )

]2

≥∫Vh2 dm −m(V )2

∫Vh2 dm

m(V )

=

∫Vh2 dm−m(V )

∫Vh2 dm = m(U)

∫Dh2 dm .


Let us begin with the following example, which will prove to be suf-ficient for our purpose.

Lemma V.4.1.3 A Poincare inequality I(D,ϕ, ψ) holds on a convexsimplex D = Da :=

(x1, . . . , xd) ∈ (R∗+)d

∣∣x1 + · · ·+ xd < a

, for all ϕand ψ (and a > 0).

Proof Fix a constant c <∞ such that ϕ ≤ c on D. We first have :

2

∫D

ϕ×∫D

f 2 ϕ− 2

(∫D

f ϕ

)2

=

∫D2

|f(x)− f(x′)|2 ϕ(x)ϕ(x′) dx dx′.

Then by the hypothesis on D, for any given x, x′ ∈ D we can find apiecewise linear path x = x0 → x1 → · · · → xd = x′ in D such thatup to some permutation

(depending on (x, x′)

)of the d coordinates, we

have xj ≡ (x′1, . . . , x′j, xj+1, . . . , xd) for 0 ≤ j ≤ d , so that each segment

[xj−1, xj] is directed by the coordinate axis (xj). Therefore

f(x)− f(x′) =d∑j=1

[f(xj−1)− f(xj)

]

=d∑j=1

∫ x′j

xj

∂jf(x′1, . . . , x

′j−1 , ξj , xj+1, . . . , xd

)dξj

and

|f(x)− f(x′)|2 ≤ dd∑j=1

(x′j − xj)∫ x′j

xj

∣∣∂jf( . . . , x′j−1 , ξj , xj+1, . . .)∣∣2 dξj

≤ d× diam(D)d∑j=1

∫ [1D ×

∣∣∂jf ∣∣2]( . . . , x′j−1 , ξj , xj+1, . . .)dξj .


Setting k := d× diam(D)× c2 , we thus have :∫D2

|f(x)− f(x′)|2 ϕ(x)ϕ(x′) dx dx′ ≤ c2

∫D2

|f(x)− f(x′)|2 dx dx′

≤ k

∫ d∑j=1

[1D∣∣∂jf ∣∣2](. . ., x′j−1, ξj, xj+1, . . .

)1D2(x, x′) dξj dx1 . . . dxd dx

′1 . . . dx

′d

≤ k diam(D)d+1

∫ d∑j=1

[1D∣∣∂jf ∣∣2](. . ., x′j−1, ξj, xj+1, . . .

)dx′1 . . . dx

′j−1 dξj dxj+1. . .dxd

= k′∫D

|df |2(ξ) dξ ≤ k′

inf ψ(D)

∫D

|df |2(ξ)ψ(ξ) dξ ,

with k′ := d× diam(D)d+2 × c2 = k′(D, c). Finally we obtain :∫D

f 2 ϕ ≤ k′

2∫Dϕ×inf

Dψ

∫D

|df |2 ψ + 1∫Dϕ

(∫D

f ϕ

)2

.

Remark V.4.1.4 The proof of Lemma V.4.1.3 is valid as well for a

parallelepipedd∏j=1

]aj, a′j[ or an open ball in Rd, and actually for any

bounded domain D ⊂ Rd for which there exists N(D) ∈ N such that anypair of points x, x′ ⊂ D can be linked by a piecewise linear path madeof at most N(D) segments (contained in D) directed by the coordinateaxes.

We then have the following invariance under diffeomorphisms.

Proposition V.4.1.5 Consider bounded domains D ,D′ in Rd, withclosures D , D′ having some C1-diffeomorphic neighbourhoods, and suchthat Poincare inequalities I(D,ϕ, ψ) hold for any ϕ, ψ as in DefinitionV.4.1.1. Then a Poincare inequality I(D′, ϕ, ψ) holds on D′ as well,for all ϕ and ψ.


Proof Let H : D → D′ be a C1 diffeomorphism (from a neighborhoodof D onto a neighborhood of D′). Note that the Jacobian det JH doesnot vanish on D, and then is bounded and bounded away from 0 on D′.Consider ϕ, ψ (as in Definition V.4.1.1) on D′, and f ∈ C1∩L1(D′). Seth := f H , Φ := ϕH×| det JH | , and Ψ := ψH×| det JH |×‖JH‖−2 .

Applying the hypothesis I(D,Φ,Ψ) to h(under the form (V.2)

)yields :∫

D′f 2 ϕ =

∫D

h2 Φ ≤ C

∫D

|dh|2 Ψ + C

(∫D

hΦ

)2

≤ C

∫D

|df |2H×‖JH‖2 Ψ +C

(∫D

hΦ

)2

= C

∫D′|df |2 ψ+C

(∫D′f ϕ

)2

.

It will be crucial to be able to consider separately different domains,in order to we obtain a Poincare inequality on their union. This will bepossible by the following. Note that we somewhat relax the constraintson the functions ϕ, ψ here, in order to apply this statement, below,not only for proving Proposition V.4.1.8, but also to the slightly moregeneral context of Theorem V.4.2.5.

Proposition V.4.1.6 Let D and D′ be two intersecting domains inRd and ϕ, ψ be two positive measurable functions on D ∪D′, such thatϕ ∈ L1(D ∪ D′). If Poincare inequalities I(D,ϕ, ψ) and I(D′, ϕ, ψ)hold, then a Poincare inequality I(D ∪D′, ϕ, ψ) holds too.

Proof Denote by µ the absolutely continuous measure with densityϕ , restricted to D ∪ D′ and normalized to be a probability measure :

µ(dx) := 1D∪D′(x)ϕ(x) dx/∫

D∪D′ ϕ , and consider any function f ∈C1 ∩ L1(D ∪D′, µ).

Let µD(f) := µ(D)−1

∫D

f dµ and VD(f) := µD(f 2)− µD(f)2 denote

respectively the mean and the variance of f on D , and write µ(f), V (f)simply for µD∪D′(f), VD∪D′(f).


Applying Definition V.4.1.1 to(f − µD(f)

)on D and to

(f − µD′(f)

)on D′, we first we obtain :

2

∫D∪D′

|df |2 ψ dx ≥∫D

|df |2 ψ dx+

∫D′|df |2 ψ dx

≥ C−1D µ(D)VD(f) + C−1

D′ µ(D′)VD′(f) . (V.4)

Set then D1 := D \D′, D2 := D′ \D, D3 := D ∩D′, and consider theconditional expectation f of f with respect to the partition D ∪D′ =

3⊔j=1

Dj : f :=3∑j=1

µDj(f)1Dj

. We have µ(f) = µ(f) =3∑j=1

µ(Dj)µDj(f),

and :V (f) = V (f − f) + V (f)

= V

( 3∑j=1

(f1Dj

− µDj(f))1Dj

)+ V

( 3∑j=1

µDj(f)1Dj

)

= µ

[ 3∑j=1

(f − µDj

(f))2

1Dj

]+ µ

[ 3∑j=1

(µDj

(f)− µ(f))2

1Dj

]

=3∑j=1

µ(Dj)[VDj

(f) +(µDj

(f)− µ(f))2].

From this expression for V (f) we obtain then :

V (f)−3∑j=1

µ(Dj)VDj(f) =

3∑j=1

µ(Dj)(µDj

(f)−µD3(f) +µD3

(f)−µ(f))2

=∑j

µ(Dj)(µDj

(f)− µD3(f))2 −

(µD3

(f)− µ(f))2

(since

∑j

µ(Dj)(µDj

(f)− µD3(f))

= µ(f)− µD3(f) and

∑j

µ(Dj) = 1)


≤∑j

µ(Dj)(µDj

(f)− µD3(f))2. (V.5)

Moreover, writing D = D1 tD3 and D′ = D2 tD3 , we have :

µ(D)VD(f) =

µ(D1)[VD1

(f)+(µD1

(f)−µD(f))2]+µ(D3)

[VD3

(f)+(µD3

(f)−µD(f))2]

= µ(D1)VD1(f) +µ(D3)VD3

(f) +µ(D)−1µ(D1)µ(D3)(µD1

(f)−µD3(f))2,

since

µ(D) = µ(D1) + µ(D3) and µ(D)µD(f) = µ(D1)µD1(f) + µ(D3)µD3

(f)

implyµ(D)

(µD1

(f)− µD(f))

= µ(D3)(µD1

(f)− µD3(f))

and the same with D1, D3 exchanged. In the same way, we have :

µ(D′)VD′(f) =

µ(D2)VD2(f) + µ(D3)VD3

(f) + µ(D′)−1µ(D2)µ(D3)(µD2

(f)− µD3(f))2.

From these two expressions, we obtain that∑j

µ(Dj)VDj(f) ≤ µ(D)VD(f) + µ(D′)VD′(f) (V.6)

and(setting C := maxµ(D), µ(D′)

/µ(D3)

)∑j

µ(Dj)(µDj

(f)− µD3(f))2 ≤ C

[µ(D)VD(f) + µ(D′)VD′(f)

]. (V.7)

The conclusion follows now at once from the above inequalities (V.4),(V.5), (V.6), (V.7) :

V (f) ≤ (C + 1)[µ(D)VD(f) + µ(D′)VD′(f)

]≤ C ′

∫D∪D′

|df |2 ψ dx ,

which is a Poincare inequality I(D ∪D′, ϕ, ψ).


Remark V.4.1.7 Some more care in the above proof yields the following estimate for thebest constant in the Poincare inequality under consideration :

CD∪D′ ≤ 2maxCD, CD′µ(D ∩D′)

.

Indeed, a careful computation yields

µ(D3)−1[µ(D)VD(f) + µ(D′)VD′(f)

]− V (f)− VD3(f)−

(µD3(f)− µ(f)

)2=[

1µ(D3) − 1

]∑j

µ(Dj)VDj (f) + µ(D1)µ(D2)[

[µD1(f)−µD3

(f)]2

µ(D) +[µD2

(f)−µD3(f)]2

µ(D′)

]≥ 0 ,

whence

V (f) ≤ µ(D)VD(f) + µ(D′)VD′(f)

µ(D3)≤ 2

maxCD, CD′µ(D ∩D′)

∫D∪D′

|df |2 ψ dx .

Proposition V.4.1.8 A Poincare inequality I(D,ϕ, ψ) holds on a do-main D ⊂ Rd whose closure is C1-diffeomorphic to a convex polyhedron,for all ϕ and ψ.

Proof By Proposition V.4.1.5 we just have to consider the case of aconvex polyhedron. Now this is a finite union of adjacent simplexes, C1-diffeomorphic to the simplexes considered in Lemma V.4.1.3. Moreoverwe obtain an overlap between adjacent simplexes by means of diffeomor-phisms acting non trivially in a small neighbourhood of their adjacentfaces. We conclude by applying Propositions V.4.1.5 and V.4.1.6.

V.4.2 The case of a fundamental domain in Hd

We apply now the proposition V.4.1.8 to our main concern : hyper-bolic domains, i.e., connected open subsets D ⊂ Hd, although most ofthe following is valid in a more general setting. Any hyperbolic domainD is endowed with the restriction of the volume measure dp

(recall


Definition III.6.2.1 and Proposition III.6.2.2(i)), and with a hyperbolic

gradient, as it appears implicitely for example in Proposition III.2.3.

Namely, to any f ∈ C2(D) we associate its squared hyperbolic gradient :

γ(f) := 12 ∆(f 2)− f ∆f = y2

(∂f

∂y

)2

+ y2d∑j=2

(∂f

∂xj

)2

, (V.8)

in Poincare coordinates (recall Proposition I.6.1), according to Formula(III.10). We are henceforth concerned with the following notion ofPoincare inequality.

Definition V.4.2.1 A Poincare inequality holds on a hyperbolic do-main D when there exists CD > 0 such that, for any integrable function

f of class C1 on D such that

∫D

f(p) dp = 0 , we have :∫D

f 2(p) dp ≤ CD

∫D

γ(f)(p) dp . (V.9)

Using Poincare coordinates, Proposition III.6.2.2(i) and (V.8), this

reads : ∫D

f 2(x, y) y−d dx dy ≤ CD

∫D

|df |2(x, y) y2−d dx dy (V.10)

as soon as

∫D

f(x, y) y−d dx dy = 0 .

This explains why we needed to consider weights ϕ, ψ in the precedingsection V.4.1. Actually we particularize now to ϕ ≡ y−d and ψ ≡ y2−d.

Applying Proposition V.4.1.8 to a bounded hyperbolic domain D, weobtain the following.


Corollary V.4.2.2 A Poincare inequality (V.9) holds on any boun-ded domain D ⊂ Hd, whose closure is C1-diffeomorphic to a convexpolyhedron.

To handle unbounded fundamental domains, we need in particular toadress their unbounded ends. We call solid cusp the image in the orbitspace Γ\Hd of a horoball H+ based at some parabolic point η , and smallenough in order that Γ\H+ can be identified with Γη\H+, where Γηdenotes the maximal parabolic subgroup of Γ fixing η (see Section IX.4).We shall only need to consider the case of an isolated solid cusp havingfinite volume, and then maximal rank, which means that the stabiliserΓη contains a free Abelian subgroup (of finite index) isomorphic to Zd−1.Thus we need to handle finite quotients (in dimension d ≥ 3 only), asfollows.

Lemma V.4.2.3 Consider the disjoint union D = D1t . . .tDk ⊂ Hd

of k bounded isometric domains D1, . . . , Dk . If a Poincare inequality(V.9) holds on D, then a Poincare inequality (V.9) holds on D1 as well.

Proof Denote by ϕj an isometry from Dj onto D1 , and consider f ∈ C1(D1), such that∫D1

f = 0 . Extending it to D by setting f := f ϕj on Dj , we first have

∫Df =

k∑j=1

∫Dj

f ϕj = k

∫D1

f = 0 , since any isometry preserves the volume measure.

Therefore (and since the Jacobian matrix Jϕj is bounded on Dj) we have :

∫D1

f2 = k−1

∫Df2 ≤ CD

k

∫Dγ(f ) = CD

k

k∑j=1

∫Dj

γ(f ϕj)

≤ CDk

k∑j=1

∫D1

γ(f)×(‖Jϕj‖2 ϕ−1

j

)≤ CD1

∫D1

γ(f) .

We come now to the Poincare inequality relative to a solid cusp.


Proposition V.4.2.4 A Poincare inequality (V.9) holds on any solidcusp of finite volume.

Proof 1) Fix a solid cusp D , and let us use a Poincare modelRd−1× R∗+ such that the cusp associated to D is ∞, and, accordingto Theorem IX.4.5, D = P×]1,∞[ for some P ⊂ Rd−1, a relativelycompact fundamental domain of the maximal parabolic subgroup Γ∞(stabilizing the cusp ∞ and containing a finite index subgroup isomor-phic to Zd−1) acting on Rd−1. We must establish (V.10), or equivalently :∫D

∣∣∣∣f(x, y)−∫Df(x, y)y−ddxdy∫

D y−ddxdy

∣∣∣∣2y−ddxdy ≤ CD

∫D

|df |2(x, y)y2−ddxdy,

for any function f of class C1 on D such that

∫D

f 2(x, y)y−ddxdy <∞ .

2) Let us first consider functions of the vertical coordinate y alone,and more precisely the space H of real C1 functions h on [1,∞[ suchthat ∫ ∞

1

h(y) y−ddy = 0 and

‖h‖2 := (d− 1)2

∫ ∞1

h(y)2 y−ddy +

∫ ∞1

|h′(y)|2 y2−ddy <∞ .

Set also H′ := h ∈ H |h(1) = 0, and ϕ(y) := (d− 1)2 log y − 1 .

It is easily checked that the function ϕ belongs to H . Furthermore,it is orthogonal to the space H′ with respect to the underlying scalarproduct, since for any h ∈ H′, integrating by parts we obtain :

(d−1)−2

∫ ∞1

h′(y)ϕ′(y)y2−ddy =[h(y) y1−d

]∞1−(1−d)

∫ ∞1

h(y)y−ddy = 0 .(The boundary term vanishes since for any h ∈ H′, Y ≥ 1 and ε ∈ ]0, 1

4 ]we have :∣∣h(Y )Y 1+ε−d∣∣2 =

∣∣∣(1+ε−d)

∫ Y

1

h(y)yε−ddy+

∫ Y

1

h′(y)y1+ε−ddy∣∣∣2≤ c‖h‖2


by Schwarz’ inequality.)

Set then

cd :=

∫ ∞1

ϕ(y)2 y−ddy

/∫ ∞1

[ϕ′(y)]2 y2−ddy .

Integrating by parts and applying Schwarz’ inequality, for any h ∈ H′we obtain :

(d− 1)

∫ ∞1

h2(y) y−ddy ≤ 2

∫ ∞1

h(y)h′(y) y1−ddy

≤ 2

√∫ ∞1

h2(y) y−ddy

∫ ∞1

[h′(y)]2 y2−ddy

and therefore

(d− 1)2

∫ ∞1

h2(y) y−ddy ≤ 4

∫ ∞1

[h′(y)]2 y2−ddy .

We obtain thus for any h ∈ H′ and t ∈ R :∫ ∞1

[h(y) + t ϕ(y)

]2y−ddy ≤ 2

∫ ∞1

h(y)2 y−ddy + 2 t2∫ ∞

1

ϕ(y)2 y−ddy

≤ 8(d− 1)−2

∫ ∞1

[h′(y)]2 y2−ddy + 2cd t2

∫ ∞1

[ϕ′(y)]2 y2−ddy

≤ c′d

∫ ∞1

[h′ + t ϕ′

]2(y) y2−ddy ,

which shows that the Poincare inequality holds in H .

3) Fix now a function f ∈ C1(D) which is square integrable with

respect to the volume measure of D :

∫D

f(x, y)2 y−ddx dy <∞ ,

denote by m its average with respect to the normalized volume measure :


m :=

∫D f(x, y) y−ddx dy∫

D y−d dx dy

, and set for any y ≥ 1 :

h(y) :=

∫P f(x, y) dx∫

P dxand H(y) := h(y)−m.

Note that (d− 1)

∫ ∞1

h(y) y−ddy = m ,

∫ ∞1

H(y)2 y−ddy <∞ and∫ ∞1

H(y) y−ddy = 0 , so that we can apply the Poincare inequality ob-

tained in 2) above to H. Moreover, by Theorem IX.4.5, the boundedpolyhedron P , base of the solid cusp D, is a quotient (by a finite groupof isometries) of a bounded polyhedron P ′ diffeomorphic to a hyper-cube. Hence by Proposition V.4.2.2 and Lemma V.4.2.3, we have forany y ≥ 1 :∫

P

[f(x, y)− h(y)

]2dx ≤ CP

∫P

∣∣dxf(x, y)∣∣2dx for some CP > 0 .

Finally, we obtain :∫D

∣∣f(x, y)−m∣∣2y−ddx dy =

∫D

([f(x, y)− h(y)

]+H(y)

)2

y−ddx dy

≤ 2

∫ ∞1

∫P

[f(x, y)− h(y)

]2dx y−ddy + 2

∫Pdx

∫ ∞1

H(y)2 y−ddy

≤ 2CP

∫ ∞1

∫P

∣∣dxf(x, y)∣∣2dx y−ddy + 2 c′d

∫Pdx

∫ ∞1

∣∣H ′(y)∣∣2 y2−ddy

≤ 2CP

∫D

∣∣dxf(x, y)∣∣2y2−ddxdy + 2 c′d

∫D

∣∣dyf(x, y)∣∣2 y2−ddxdy ,

whence the Poincare inequality on D, with CD := 2 maxCP , c′d.

We can now conclude, with the main result of the present section V.4,which yields the spectral gap result we need for Chapter VIII.


Figure V.1: ideal core-cusp decomposition for Theorem V.4.2.5

Theorem V.4.2.5 A Poincare inequality (V.9) holds on a fundamentaldomain D of a geometrically finite and cofinite Kleinian group Γ.

Proof We use Corollary V.4.2.2, and Propositions V.4.2.4 and V.4.1.6.For this, we note that in the cofinite examples of Chapter IV, we cancover a fundamental polyhedron by a finite union of solid cusps and a so-called core, which is compact and diffeomorphic to a convex polyhedron.

Indeed, the solid cusps are obtained by cutting each of the unboundedends of the fundamental polyhedron by a sufficiently small horoball(based at the corresponding infinite end, i.e., cusp). The remainingcore is diffeomorphic to a convex polyhedron. Then this core can beslightly enlarged, by cutting each unbounded end by a strictly smallerhoroball, in order to intersect the interior of every solid cusp (and to bestill diffeomorphic to a convex polyhedron). See Figures V.1 and V.2.

Finally, the core-cusp decomposition of Theorem IX.4.5 (again enlargingslightly the core, to let it intersect the solid cups) states that the abovepicture holds generally.

V.5. NOTES AND COMMENTS 165

Figure V.2: non-ideal core-cusp decomposition for Theorem V.4.2.5

V.5 Notes and comments

The ergodic theorem on hyperbolic surfaces is due to Hopf [Hop]. Afundamental reference on this subject is [CFS]. See also [Rag], [Ni], [Mar],[Gu], [HK], [Da]. Otherwise [N2] and [St] prove the basic subadditiveergodic theorem elegantly.

The ergodic theorem extends to general manifolds with negative cur-vature and finite volume. In infinite volume and constant negative curva-ture, an invariant measure can be defined, namely the Patterson-Sullivanmeasure, which is related to the invariant measure of the image of theLaplacian under the ground state transformation : f 7→ 1

h ∆(hf). See inparticular [P1], [P2], [PS], [Su], [Y].

The spectral gap property (equivalenty, a Poincare inequality) has animportant meaning in quantum mechanics, as it shows the existence ofa first excited state for the Hamiltonian associated with the energy.

Proposition V.4.1.2 means that a spectral gap I(D,ψ, ψ) yields a positive bottomfor the spectrum of the induced theory obtained by a so-called killing procedure on a


non-empty open set U ⊂ D(provided ϕ ∈ L1(D)

). It can be applied as follows : let

TU = infs ≥ 0 | zs ∈ U be the hitting time of U by the drifted (by ∇ logψ) Brow-nian motion (zs), and (QU

s ) be the so-called killed semi-group, defined on Cb(D) by :QUs f(z) := Ez

[f(zs) 1TU>t

]. It can be shown that the above positivity of the bottom

of the spectrum entails an exponential decay of (QUs ), which applied in particular to the

constant function f ≡ 1 yields an exponential decay of the tail P(TU > s) as s→∞ ,or equivalently, some exponential integrability of the hitting time TU . See also [CK].

Poincare inequalities pertain to a family of functional inequalitieswhich have been extensively developed in the recent years, in particularLog-Sobolev inequalities (due to L. Gross). See for example [By].

Chapter VI

Basic Ito Calculus

The passage from the discrete to the continuum is fundamental inprobability theory. It originates with De Moivre’s Theorem on the con-vergence of the rescaled binomial distribution towards the bell-shapedcurve. But it is only during the XX-th century that the scaling limit ofa simple random walk, known as Wiener process or Brownian motion,was mathematically defined and studied. Ito’s calculus is the continu-ous limit of elementary calculations that can be done in discrete time. Itturns out that a second order correction to the usual rules of calculus areneeded, arising from the fact that Brownian paths have a non-vanishingquadratic variation.

The fundamental notions such as predictability, martingales, stoppingtimes, are introduced in Section 1 in the discrete case. We then give ashort introduction to Brownian motion and finally deal with the basictools of Ito’s calculus : the stochastic integral and the Ito change ofvariable formula. We keep trying to avoid unnecessary technicalities,although providing complete proofs.

167

168 CHAPTER VI. BASIC ITO CALCULUS

VI.1 Discrete martingales and stochastic integrals

We assume that the reader is familiar with the notion of probabilityspace. Random variables will always be defined almost surely.

Consider a probability space (Ω,F ,P), endowed with a filtration (Fn),i.e., an increasing sequence of sub-σ-fields of F , which will often begenerated by a process.

Definition VI.1.1 An (Fn)-martingale is a sequence of real randomvariables Xn |n ∈ N such that for any n ∈ N :

i) Xn is Fn-measurable (defining a non anticipating sequence) andintegrable : Xn ∈ L1(Fn ,P);

ii) Xn = E(Xn+1|Fn) almost surely.

If in ii), = is replaced by ≤ or by ≥ , the process (Xn) is said to beinstead a submartingale or a supermartingale, respectively.

Note that for a submartingale, E(Xn) is non-decreasing and that for asupermartingale, E(Xn) is non-increasing.

Examples : - If Z is integrable, then E(Z|Fn) defines a martingale.

- If Sn :=n∑j=1

Xj denotes a simple random walk on Z, with independent

variables (Xj) such that P(Xj = 1) = p = 1−P(Xj = −1), then this isa martingale if and only if p = 1

2 , a submartingale if and only if p ≥ 12 ,

and a supermartingale if and only if p ≤ 12 .

By Jensen’s inequality for conditional expectations, if (Xn) is a mar-tingale and φ a convex real function such that E

(|φ Xn|

)<∞ , then

(φ Xn) is a submartingale.

Moreover, if (Xn) is a submartingale and φ a convex non-decreasing realfunction such that E

(|φ Xn|

)< ∞ , then (φ Xn) is a submartin-

VI.1. DISCRETE MARTINGALES AND STOCHASTIC INTEGRALS 169

gale. For example, (Xn − a)+ and maxXn, a are submartingales, forany real constant a . Consequently, if (Xn) is a supermartingale, thenminXn, a is a supermartingale.

Definition VI.1.2 A predictable process in discrete time is a processHn |n ∈ N∗ such that Xn is Fn−1-measurable for all n ≥ 1.

Think of Hn as the amount a gambler bets on the n-th game, knowingthe outcome of the previous games.

Proposition VI.1.3 Let H be bounded and predictable and X be a

martingale, and define (H ·X)n :=n∑

m=1Hm(Xm −Xm−1). Then H ·X

is a martingale. If H is non-negative bounded and predictable and if Xis a sub or supermartingale, then so is H ·X .

This is an essential property. H ·X is called the stochastic integral ofH against X . The proof is straightforward from the definitions.

Recall that an (Fn)-stopping time is a random variable S taking val-ues in N = N ∪ ∞ such that S = n ∈ Fn for all n . (e.g., hittingtimes for random walks, or more generally for any non-anticipating pro-cess.)

The process Hn := 1S≥n = 1−n−1∑k=0

1S=k is predictable and

(H · X)n = Xn∧S − X0 . Moreover, ((1 − H) · X)n = Xn − Xn∧S .Therefore, we have the following.

Corollary VI.1.4 If X is a submartingale, then Xn∧S |n ∈ N andXn−Xn∧S |n ∈ N are submartingales too. In particular, E(XS∧n) ≤E(Xn) for any n ∈ N .


Note the last point can be seen directly : indeed,

E(XS∧n) =n∑k=0

E(Xk 1S∧n=k

)≤

n∑k=0

E(E(Xn|Fk) 1S∧n=k

)= E(Xn).

Example : Doubling strategy, in a coin tossing game : double your betsuntil you win ! Mathematically, this is modelled by a simple random walk(Sn). Set τ := infn |Xn = 1, and take Hn := 2n−11τ≥n. You win 1at time τ . Beware that, in practice, you cannot play that way infinitelymany times, so your risk to loose a lot is significant.

As a consequence of the previous corollary, we get the following.

Theorem VI.1.5 (Doob’s inequalities) Let (Xn) be a submartingale.Then :

(i) for any λ > 0 , we have :

λ P[

supm≤n

X+m ≥ λ

]≤ E

[Xn 1

supm≤n

X+m≥λ

]≤ E(X+

n ) ;

(ii) if (Xn) is non-negative or is a martingale, then for any p > 1 :∥∥∥ supm≤n|Xm|

∥∥∥p≤ p

p− 1‖Xn‖p for all n ,

and ∥∥∥ supn|Xn|

∥∥∥p≤ p

p− 1supn‖Xn‖p .

Proof (i) For the first inequality, denoting by T the hitting time of

[λ , ∞[ by X , we have : T ≤ n =

supm≤n

X+m ≥ λ

= XT∧n ≥ λ,

and then :

λP[

supm≤n

X+m ≥ λ

]≤ E

[XT∧n1T≤n

]=

n∑k=1

E[Xk 1T=k

]≤ E

[Xn 1T≤n

].

VI.1. DISCRETE MARTINGALES AND STOCHASTIC INTEGRALS 171

The second inequality is obvious.

(ii) The absolute value of a martingale is a non-negative submartin-gale. It is thus enough to consider a non-negative submartingale (Xn).Note that ‖Xn‖p increases, since Xp is also a submartingale. Set

Yn := supm≤n

Xm . From (i), by Fubini’s Theorem we get :

‖Yn‖pp =

∫ ∞0

p λp−1 P(Yn ≥ λ) dλ ≤ p

∫ ∞0

λp−2 E[Xn1Yn≥λ

]dλ

= pE[Xn

∫ Yn

0

λp−2 dλ

]=

p

p− 1E(Xn Y

p−1n

)≤ p

p− 1‖Xn‖p ‖Y p−1

n ‖ pp−1

=p

p− 1‖Xn‖p ‖Yn‖p−1

p

by Holder’s inequality. This yields the result if 0 < ‖Yn‖p <∞ . Thereis nothing to prove if ‖Yn‖p = 0 , and if ‖Yn‖p =∞ , then we must have‖Xn‖p = ∞ too, since ‖Yn‖p ≤ ‖X0‖p + . . . + ‖Xn‖p ≤ (n + 1)‖Xn‖p .

Finally, this implies at once the inequality relating to

∥∥∥∥supnYn

∥∥∥∥p

, merely

by applying the monotone convergence theorem. A martingale (Mn) which is bounded in L2 is easily seen to converge

in L2 : indeed, since its L2 squared norm is the sum of the squared normsof its increments :

n∑k=1

‖Mk −Mk−1‖22 = ‖Mn‖2

2 − ‖M0‖22 ≤ C <∞ ,

the Cauchy criterion applies at once. Letting m go to infinity in

E(Mn+m | Fn) = Mn , we deduce that E(M∞ | Fn) = Mn for any n ,almost surely.


VI.2 Brownian Motion

In continuous time, we will consider only continuous or cadlag (i.e.,left limited and right continuous) matrix valued processes, i.e., randomvariables taking values in the space of continuous or cadlag functionsXt of the time coordinate t ∈ R+ , equipped with its natural σ-field :σXs | s ∈ R+ = σXs | s ∈ D, D denoting the set of non-negativedyadic numbers.

Definition VI.2.1 A real Brownian motion (or Wiener process) is areal valued continuous process (Bt)t≥0 such that for any n ∈ N∗ and0 = t0 < · · · < tn , the random variables (Btj − Btj+1

) are independent,and the law of (Btj − Btj+1

) is N (0, tj − tj−1) , i.e., centred Gaussianwith variance (tj+1 − tj).

The above Brownian motion (Bt) is sometimes called standard to underline the fact that it

starts from 0 and has unit variance (at time 1). The generic real Brownian motion has the

law of (a+ cBt), for real constants a, c .

A slightly different formulation of the second part of the definition is :

the increments of (Bt) are independent, and stationary : s ≤ t ∈ R∗+,

(Bt −Bs)law≡ Bt−s , and the law of Bt is N (0, t).

A simple construction of (Bt), by means of a multi-scale series, isgiven in the appendix, see Section X.1.

The (probability) law of such a process is clearly unique, and is knownas the Wiener measure on the space C0(R+,R) of real continuous func-tions indexed by R+ (and vanishing at 0).

The following property is straightforward from the definition, sincethe law of a Gaussian process is prescribed by its mean and its covariance.

VI.2. BROWNIAN MOTION 173

Proposition VI.2.2 The standard real Brownian motion (Bt) is theunique real process which is Gaussian centred with covariance functionR2

+ 3 (s, t) 7−→ E(BsBt) = mins, t.

The processes t 7→ Ba+t − Ba , t 7→ c−1Bc2t , t 7→ t B1/t , and t 7→(BT − BT−t) (for 0 ≤ t ≤ T ) satisfy the same conditions. We thereforededuce the following fundamental properties :

Corollary VI.2.3 The standard real Brownian motion (Bt) satisfies

1) the Markov property : for all a ∈ R+ , (Ba+t−Ba) is also a standardBrownian motion, and is independent from Fa := σBs | 0 ≤ s ≤ a ;

2) the self-similarity : for any c > 0, (c−1Bc2t) is also a standard realBrownian motion ;

3) (−Bt) and (t B1/t) are also standard real Brownian motions.

4) for any fixed T > 0 , (BT − BT−t)0≤t≤T is also a standard realBrownian motion (on [0, T ]).

An Rd-valued process(B1t , . . . , B

dt

)made of d independent standard

Brownian motions (Bjt ) is called a d-dimensional Brownian motion.

Of course, for any v ∈ Rd, v+(B1t , . . . , B

dt

)is also called a d-dimensional

Brownian motion, starting at v .

Exercise Prove that the law of a d-dimensional Brownian motion (start-ing at 0) is preserved by Euclidean rotations

(of the vector space Rd

).

Proposition VI.2.4 (quadratic variation) Fix t > 0 , and denote byP := 0 = t0 < t1 < . . . < tN = t a subdivision of [0, t] , with mesh |P| .Set VP =

N−1∑j=0

(Btj+1− Btj)

2 , for some standard real Brownian motion

(Bt). Then as |P| goes to 0, VP converges to t in L2-norm.


Proof We have E(VP) = t , and by the independence of increments :

E[(VP − t)2

]=

N−1∑j=0

(tj+1 − tj)2 × E([

(Btj+1−Btj )2

tj+1−tj − 1]2)

= 3N−1∑j=0

(tj+1 − tj)2 ≤ 3t |P| .

VI.3 Martingales in continuous time

Consider a probability space (Ω,F ,P) endowed with a filtration (Ft),i.e., an increasing family of sub-σ-fields of F . We suppose that F0 con-tains the set N (P) of all P-negligible subsets of Ω .

Definition VI.3.1 A continuous (Ft)-martingale is a continuous realvalued process Xt | t ∈ R+ such that :

i) Xt is Ft-measurable (i.e., a non-anticipating or adapted process) andintegrable : Xt ∈ L1(Ft ,P);

ii) Xt = E(Xt+s | Ft) almost surely, for all s, t ≥ 0 .

If in ii), = is replaced by ≤ or by ≥ , the process (Xt) is said to beinstead a submartingale or a supermartingale, respectively.

The properties of discrete martingales extend to martingales in contin-uous time. In particular if (Xt) is a submartingale, then (Xt − a)+ is asubmartingale. If (Xt) is a supermartingale, then (t 7→ minXt , a) isa supermartingale.

Examples: - A Brownian motion (Bt) is a square integrable continuousmartingale, with respect to its completed proper filtration

F t = σBs | s ≤ t ∨ N (P). This holds for (B2t − t) as well.

VI.3. MARTINGALES IN CONTINUOUS TIME 175

- For any real α , the “exponential martingale” (eαBt−α2 t/2) is indeed a

(continuous) martingale.

The following is easily deduced from Theorem VI.1.5, by using dyadicapproximation.

Theorem VI.3.2 (Doob’s inequalities) Let (Xt) be a continuous sub-martingale. Then :

(i) for any λ, t > 0 , we have :

λ P[

sups≤t

X+s ≥ λ

]≤ E

[Xt 1

sups≤t

X+s ≥λ

]≤ E(X+

t ) ;

(ii) if (Xt) is non-negative or is a martingale, then for any p > 1 :∥∥∥ sups≤t|Xs|

∥∥∥p≤ p

p− 1‖Xt‖p for any time t ,

and then ∥∥∥ sups|Xs|

∥∥∥p≤ p

p− 1sups‖Xs‖p .

We shall use the following asymptotic property, known as Khintchinelaw of iterated logarithm. The elegant proof we give (taken from [MK])uses essentially Theorem VI.3.2(i), applied to the exponential martingalegiven above as an example.

Proposition VI.3.3 For any real Brownian motion (Bt), we almost

surely have lim supt→∞

Bt√2t log(log t)

= 1 = − lim inft→∞

Bt√2t log(log t)

.

Proof We have to prove that lim supt 0

Bt√2t log log(1/t)

= 1 . This is

indeed equivalent to the claim via the transforms (Bt) 7→ (−Bt) and(Bt) 7→ (t B1/t) of Corollary VI.2.3(3).


Set h(t) :=√

2t log log(1/t) , and fix θ, δ ∈ ]0, 1[ .

Set also αn := (1 + δ) θ−n h(θn) and βn := h(θn)/2 , for any n ∈ N∗.Doob’s inequality (precisely Theorem VI.3.2(i)) applied to the exponen-tial martingale Xs := eαnBs−α

2ns/2 ensures that

P[

sup0≤s≤1

(Bs−αns2 ) ≥ βn

]= P

[sup

0≤s≤1Xs ≥ eαnβn

]≤ e−αnβn =

(n|log θ|

)−1−δ,

whence P(

lim infn

sup

0≤s≤1(Bs − αns/2) < βn

)= 1 by Borel-Cantelli

Lemma. Hence, we almost surely have : for any large enough n and forθn ≤ s < θn−1,

Bs ≤ βn + αns2 ≤ βn + αnθ

n−1

2 =(

12 + 1+δ

2θ

)h(θn) ≤

(12 + 1+δ

2θ

)h(s) .

This proves that almost surely lim sups 0

Bs/h(s) ≤(

12 + 1+δ

2θ

), whence

lim sups 0

Bs/h(s) ≤ 1 , letting (1− θ) and δ go to 0 .

As we shall only use this upper bound, we refer to [MK] for the analogousproof of the lower bound.

Note that the above law of iterated logarithm, under its form lim supt 0

Bt√2t log log(1/t)

= 1,

shows the almost sure non-differentiability of the Brownian path at 0, and then at any a ∈ R+

by the Markov property (of Corollary VI.2.3). Actually this non-differentiability occurs in a

stronger sense, see for example([R-Y], chapter I, (2.9)

).

Definition VI.3.4 An (Ft)-stopping time is a random variable S

taking values in R+ = R+ ∪ ∞ such that S ≤ t ∈ Ft for all t ≥ 0 .

The σ-field associated to a stopping time S is

FS :=F ∈ F

∣∣F ∩ S ≤ t ∈ Ft for any t ≥ 0

.

This implies that S < t ∈ Ft for all t , and this is actually equivalent ifthe filtration is right-continuous, which means that Fs+ :=

⋂t>sFt equals

VI.3. MARTINGALES IN CONTINUOUS TIME 177

Fs for any s > 0 . Then classical examples of stopping times are thehitting times of open sets by a right-continuous non-anticipating process.

It is easy to check that if S and T are stopping times, their supremumor infimum is also a stopping time. Almost sure limits of non-decreasingsequences of stopping times are stopping times, and the same holds fornon-increasing sequences if the filtration is right-continuous.

If S, T are stopping times then the stochastic interval

[S, T [ :=

(s, ω) ∈ R+ × Ω∣∣S(ω) ≤ s < T (ω)

is adapted and cadlag.

If S ≤ T , then FS ⊂ FT .

A stopping time can always be approximated uniformly from aboveby a non-increasing sequence of discrete valued stopping times, by takingSn = k+1

2n on k2n ≤ S < k+12n , to approach S.

This approximation procedure easily implies that the Markov prop-erty of CorollaryVI.2.3 extends to stopping times, as follows.

Corollary VI.3.5 (Strong Markov property) A Brownian motion (Bt)satisfies the strong Markov property : for any finite stopping time S,(BS+t−BS) is also a Brownian motion, and is independent from FS

(if

(Ft) denotes the natural Brownian filtration of Corollary VI.2.3, or atleast, provided for all 0 < s < t , (Bt −Bs) are independent from Fs

).

Proof This is straightforward from the weak Markov property (Corol-lary VI.2.3) if T =

∑j∈N

αj1Aj with Aj ∈ Fαj for each j . For the general

case, consider a sequence Tn of stopping times having the precedingform, which decreases to T : we get so a sequence of standard Brownianmotions, which are independent from FT and which converge

(almost

surely uniformly on any T ≤ N, t ≤ N)

to (BT+t −BT ).


Proposition VI.3.6 (Hitting Times) For all t, x > 0, the law of hit-ting time Tx := mins > 0 |Bs = x of a real Brownian motion (Bs) isgiven by :

P(Tx < t) = P(maxBs| s ≤ t > x) = 2P(Bt > x) = P(|Bt| > x)

=

∫ t

0

x√2π s3

e−x2

2s ds .

Proof The second equality is due to the reflection principle, betweentimes Tx and t , on the event Tx < t : setting T tx := minTx , t, so thatTx < t = T tx < t, and using the strong Markov property (CorollaryVI.3.5), we know that the random variable BT tx+(t−T tx) − BT tx is centredGaussian (null on T tx = t) conditionally on FT tx ; so that we have :

P(Bt > x) = P(Tx < t , Bt−BTx > 0) = P(T tx < t , BT tx+(t−T tx)−BT tx > 0

)= E

(P(BT tx+(t−T tx)−BT tx > 0

∣∣FT tx) 1T tx<t)

= E(

12 1T tx<t

)= 1

2 P(Tx < t) .

For the last equality, differentiate 2P(Bt > x) =√

2π t

∫ ∞x

e−y2

2t dy with

respect to t . NB Note in particular that the hitting times Tx are not integrable (for any x 6= 0).

Proposition VI.3.7 Denote by Mc the space of continuous squareintegrable martingales, and by M∞

c the subspace of those which arebounded in L2. Then

(i) Any (Mt) ∈ M∞c converges in L2 to some random variable M∞ ∈

L2. Moreover E(M∞ | Ft) = Mt for any t , almost surely.

(ii) M∞c is a Hilbert space for the norm ‖M‖ := ‖M∞‖2 .

VI.4. THE ITO INTEGRAL 179

Proof The discrete time case (recall the end of Section VI.1) providesa random variable M∞ ∈ L2, limit in L2 of Ms along any sequencen/2m |n ∈ N. As above, we have :∥∥M∞ −Mn/2m

∥∥2

2≤

∞∑k=n+1

∥∥Mk/2m −M(k−1)/2m∥∥2

2= ‖M∞‖2

2 −∥∥Mn/2m

∥∥2

2.

Now for any continuous martingale, continuity holds also in L2-norm, byDoob’s second inequality

(i.e., Theorem VI.1.5(ii)

). Thus we get :

‖M∞ −Mt‖22 ≤ ‖M∞‖2

2 − ‖Mt‖22 for any positive t , and the right hand

side decreases to 0 as t increases to infinity. This yields the statement(i). Moreover, the same Doob’s inequality implies also straightforwardlyStatement (ii).

Remark VI.3.8 (i) The convergence Mt →M∞ of Proposition VI.3.7occurs also almost surely. Equivalently, L2 convergence of martingalesentails almost sure convergence. See for example [Do], or [N1], or

([R-Y]

Theorem II.2.10).

(ii) If (Ms) belongs to Mc , it is clear by Corollary VI.1.4 that for anybounded stopping time T , (s 7→Ms∧T ) belongs to M∞

c .

VI.4 The Ito integral

We will now define the notion of stochastic integral in continuoustime. This is the Ito integral. The problem is to make good sense of

expressions of the form

∫ t

0

Hs dBs , where (Bs) is a Brownian motion

which is not differentiable and has infinite variation on any non-emptyopen interval. The solution is to restrict in a suitable way the class ofprocesses we integrate.


The simplest cadlag adapted processes are the step processes, i.e., theprocesses (Hs) which can be expressed as

Hs(ω) :=∞∑j=0

Uj(ω) 1[Tj ,Tj+1[(s) , (VI.1)

with a non-decreasing sequence 0 = T0 ≤ T1 ≤ . . . of stopping timesgoing almost surely to infinity, Uj ∈ bFTj (i.e., Uj is bounded and FTj-measurable), and such that E

[∫ ∞0

H2s ds

]<∞ .

Let Λ∞ denote the space of all cadlag adapted processes such that

E[∫ ∞

0

H2s ds

]<∞ . It clearly contains the step processes.

Note for future reference that the integrals of these processes are almost

surely continuous, since

∣∣∣∣∫ t

t−εHs ds

∣∣∣∣2 ≤ ε

∫ t

0

H2s ds .

Lemma VI.4.1 Step processes are dense in Λ∞ in the sense that forall H in Λ∞, there exists a sequence of step processes H(n) such that

limn→∞

∫ ∞0

E[|Hs −H(n)

s |2]ds = 0 . This holds even with simple step pro-

cesses, i.e. step processes of the form (VI.1) but with constant times Tj .

Proof Take for example H(n)s := max

− n ,minn,Hs

and

H(n)s :=

∑j≥0

H(n)

T(n)j

1[T

(n)j , T

(n)j+1[

(s) , where T(n)0 := 0 and

T(n)j+1 := inf

s > T

(n)j

∣∣ |H(n)s − H(n)

T(n)j

| > 2−n

. The second claim follows

immediately from the former and from the uniform approximation ofstopping times by discrete valued ones.


Similarly, for any t ≥ 0 let Λt denote the space of cadlag adapted

processes such that E[∫ t

0

H2s ds

]<∞ .

Assume we are given an (Ft)-Brownian motion B , that is to say, an(Ft)-adapted Brownian motion such that for all 0 < s < t , (Bt − Bs)is independent of Fs . The Ito integral is defined naturally on stepprocesses H written as in (VI.1), by :∫ t

0

H dB ≡∫ t

0

Hs dBs :=∑j≥0

Uj (BTj+1∧t −BTj∧t) .

Equivalently, denoting by N(t) the largest integer j such that Tj ≤ t

and setting UN(t) = 0 :∫ t

0

H dB =

N(t)−1∑j=0

Uj (BTj+1−BTj) + UN(t) (BT −BTN(t)

).

Lemma VI.4.2 (i)

∫ t

0

H dB is a continuous martingale ;

(ii)

∫ t

0

H dB does not depend on the representation (VI.1) of H ;

(iii) For any step process, we have for all t , positive or infinite :

E

[∣∣∣∣∫ t

0

Hs dBs

∣∣∣∣2]

= E[∫ t

0

H2s ds

].

Proof is left as an exercise.(

For (i), consider the case N(t) = 2 first

and s < T1 < t < T2 , to understand why E[∫ t

0

H dB∣∣Fs] =

∫ s

0

HdB .


(iii) follows very easily from a simple calculation, in which all crossterms vanish under the expectation, by definition of an (Ft)-Brownian

motion.)

It is then clear (using Proposition VI.3.7) that the Ito integral extendsby density to a linear map Is from Λ∞ into the space M∞

c of squareintegrable continuous martingales bounded in L2, and that we have

E[∫ ∞

0

H2s ds

]= E

[Is(H)2

∞]

=∥∥Is(H)

∥∥2.

Is(H)t is usually denoted by

∫ t

0

Hs dBs or

∫ t

0

H dB , for all t ≤ ∞ .

This notation is coherent with the following natural property.

Lemma VI.4.3 For any stopping time τ and H ∈ Λ∞, t 7→ Ht 1[0,τ [(t)belongs to Λ∞,∫ ∞

0

1s<τHs dBs = Is(H)τ =:

∫ τ

0

H dB ,

and

E

[∣∣∣∣∫ τ

0

H dB

∣∣∣∣2]

= E[∫ τ

0

H2s ds

].

Proof This is obvious for step processes, and if Hn converges towardsH in Λ∞, then Hn 1[0,τ ] also converges towards H 1[0,τ ] in Λ∞, and wehave (by Doob’s inequality) the almost sure uniform convergence of thetwo stochastic integrals.

Remark VI.4.4 In exactly the same way, we see that for any pair ofstopping times σ ≤ τ and any (Ht) in Λ∞, (t 7→ Ht 1[σ,τ [(t)) belongs toΛ∞, and∫ ∞

0

1σ≤s<τHs dBs = Is(H)τ − Is(H)σ =:

∫ τ

σ

Hs dBs ≡∫ τ

σ

H dB .


Moreover the Ito isometric identity holds :

E

[∣∣∣∣∫ τ

σ

H dB

∣∣∣∣2]

= E[∫ τ

σ

H2s ds

].

The following is known as Doob’s optional sampling theorem (writtenhere for stochastic integrals).

Proposition VI.4.5 For any stopping time T , H ∈ Λ∞ and (Ft)-Brownian motion (Bs), we have almost surely :

E[∫ ∞

0

H dB

∣∣∣∣FT] =

∫ T

0

H dB .

Proof By definition of Is(H), we have to verify that for any A ∈FT , E

[1A

∫ ∞T

H dB

]= 0 . This holds for constant times T by Lemma

VI.4.2(i). Therefore if T =∑k∈N

1Ak k2−n (with pairwise disjoint Ak ∈Fk2−n) is a discrete valued stopping time, we have :

E[1A

∫ ∞T

HdB

]=∑k∈N

E[1A

∫ ∞k2−n

1AkHdB

]=∑k∈N

E[1A∩Ak

∫ ∞k2−n

HdB

]= 0.

Hence if (Tn) is a non-increasing sequence of stopping times converginguniformly to T , we have A ∈ FTn and then :

E[1A

∫ ∞T

H dB

]= lim

n→∞E[1A

∫ ∞Tn

H dB

]= 0 .

We can extend slightly the definition of the Ito integral by consideringthe space


Λ :=⋂t>0

Λt of cadlag adapted processes such that E[∫ t

0

H2s ds

]<∞

for all t . It is obvious that the Ito integral

∫ t

0

H dB extends to a

linear map from Λ into the space Mc of square integrable continuousmartingales, and that we still have the Ito isometric identity :

E

[∣∣∣∣∫ t

0

H dB

∣∣∣∣2]

= E[∫ t

0

H2s ds

]. Note that it is equivalent to :

E[(∫ t

0

HdB

)×(∫ t

0

KdB

)]= E

[∫ t

0

HsKsds

], ∀ H,K ∈ Λ . (VI.2)

As before, for any H in Λ and for any stopping times σ ≤ τ , H 1[σ,τ [

belongs to Λ and

∫ t

0

1σ≤s<τHs dBs =

∫ t∧τ

t∧σH dB . Moreover, for any

H ∈ Λ , t ≥ 0 and any stopping times σ ≤ τ we have :

E

[∣∣∣∣∫ t∧τ

t∧σH dB

∣∣∣∣2]

= E[∫ t∧τ

t∧σH2s ds

]. (VI.3)

In particular, (t 7→ Bt∧τ) is a square integrable martingale.

Exercise Show that if E(τ) is finite, then E(Bτ) = 0 and E(B2τ ) =

E(τ). Deduce that the expectation of the hitting time of 1 by B isinfinite.

VI.5 Ito’s Formula

To prove Ito’s formula, which is the fundamental result of stochasticcalculus, we begin with two lemmas. First observe the following.

VI.5. ITO’S FORMULA 185

Lemma VI.5.1 For any cadlag adapted process H bounded by C andfor any s, t ≥ 0 , we have :

E

[∣∣∣∣∫ t

s

H dB

∣∣∣∣4]≤ 9C4(t− s)2.

Proof We can take s = 0 for simplicity. Let us first consider simplestep processes. If Ht(ω) =

∑j≥0

Uj(ω) 1]tj ,tj+1](t), with maxj|Uj| ≤ C ,

then we have :

E

[∣∣∣∣∫ t

0

H dB

∣∣∣∣4]

= E

[∑j≥0

U 4j (Bt∧tj+1

−Bt∧tj)4

]

+ 6 E

∑0≤j<k<N

U 2j U

2k (Bt∧tj+1

−Bt∧tj)2(Bt∧tk+1

−Bt∧tk)2

+12E

∑0≤i<j<k<N

UiUjU2k [Bt∧ti+1

−Bt∧ti][Bt∧tj+1−Bt∧tj ][Bt∧tk+1

−Bt∧tk]2

,as all other terms vanish. The first term is clearly bounded by 3C4 t2,and the sum of the two last terms equals :

6∑k≥1

E

(k−1∑j=0

Uj (Bt∧tj+1−Bt∧tj)

)2

U 2k (Bt∧tk+1

−Bt∧tk)2

,and then, by conditioning first with respect to Ftk, is bounded by

6C2∑k≥1

(t ∧ tk+1 − t ∧ tk)E[∣∣∣∣∫ t∧tk

0

H dB

∣∣∣∣2]≤ 6C2 t E

[∫ t

0

H2s ds

]≤ 6C4 t2.


This gives the result for simple step processes. The general case followsimmediately by Fatou’s Lemma and an approximation by a boundedsequence of simple step processes (recall Lemma VI.4.1), from which wecan extract a subsequence, to get almost sure convergence.

The following lemma is the essential step in the proof of Ito’s Formula.

Lemma VI.5.2 For any bounded cadlag adapted process H in Λ , wehave :

(i)n∑k=1

∣∣∣∣∣∫ tk

n

t(k−1)n

Hs dBs

∣∣∣∣∣2

converges to

∫ t

0

H2s ds in L2 as n→∞ ;

(ii)n∑k=1

Y t(k−1)n

∫ tkn

t(k−1)n

Hs dBs converges to

∫ t

0

YsHs dBs in L2 as n →

∞ , for any bounded continuous cadlag adapted process Y .

Proof (i) First, note that for any positive s and t,

E

[(∫ t+s

t

Hu dBu

)2

−∫ t+s

t

H2u du

∣∣∣∣Ft]

= 0 .

Indeed, for any Ft-measurable set A, τ = 1A(ω)t + 1Ac(ω)(t + s) is astopping time and Remark VI.4.4 (with σ ≡ t) shows that

E

[(∣∣∣∣∫ t+s

t

Hu dBu

∣∣∣∣2 − ∫ t+s

t

H2u du

)1A

]vanishes.

From that, it comes easily that

E n∑

k=1

∣∣∣∣∣∫ tk

n

t(k−1)n

H dB

∣∣∣∣∣2

−∫ t

0

H2s ds

2=n∑k=1

E∣∣∣∣∣

∫ tkn

t(k−1)n

H dB

∣∣∣∣∣2

−∫ tk

n

t(k−1)n

H2s ds

2,


since all cross terms expectations vanish. By Lemma VI.5.1, this quan-tity is dominated by

2n∑k=1

E

∣∣∣∣∣∫ tk

n

t(k−1)n

H dB

∣∣∣∣∣4

+

∣∣∣∣∣∫ tk

n

t(k−1)n

H2s ds

∣∣∣∣∣2 ≤ 20C4 t2/n .

(ii) Set Y n,t :=n−1∑k=0

Y t(k−1)n

1[ tkn ,t(k+1)n [ , so that

n∑k=1

Y t(k−1)n

∫ tkn

t(k−1)n

Hs dBs =

∫ t

0

Y n,ts Hs dBs .

This is indeed obvious for any simple step process H, and this extendsby continuity to every H ∈ Λ . Then (ii) follows from the convergenceof Y n,tH to Y H1[0,t[ in Λ∞, by dominated convergence.

Let us now define the space Sb (respectively S) of semimartingalesof bounded (respectively L2) type, as the space of square integrablecontinuous processes which can be written as a sum

x0 +

∫ t

0

Hs dBs +

∫ t

0

Ks ds

with x0 ∈ R, and H,K bounded cadlag adapted processes (respectivelyprocesses belonging to Λ).

Nota Bene : S is made of continuous semimartingales and is includedin Λ , since

E

[∫ t

0

∣∣∣∣∫ u

0

Ks ds

∣∣∣∣2du]≤ t2

2E[∫ t

0

K2sds

]


and

E

[∫ t

0

∣∣∣∣∫ u

0

H dB

∣∣∣∣2du]≤ t E

[∫ t

0

H2sds

].

Lemma VI.5.2 extends easily to Sb , since the treatment of the terms∫ t

0

Ks ds is straightforward, with obvious upper bounds. We thus get

the following.

Lemma VI.5.3 For any Xt = x0 +

∫ t

0

Hs dBs +

∫ t

0

Ks ds in Sb , we

have :

(i)n∑k=1

(X tkn−X t(k−1)

n)2 converges to

∫ t

0

H2s ds in L2 as n→∞ ;

(ii)n∑k=1

Y t(k−1)n

(X tkn−X t(k−1)

n) converges to

∫ t

0

YsHs dBs +

∫ t

0

YsKs ds

in L2 as n → ∞ , for any bounded continuous cadlag adapted processY .

Theorem VI.5.4 (Ito’s Formula) Consider a semimartingale in Sb :

Xt := x0 +

∫ t

0

Hs dBs +

∫ t

0

Ks ds

and a C2 function Φ having bounded derivatives. Then Φ(Xt) is in Sb ,and

Φ(Xt)−Φ(x0) =

∫ t

0

Φ′(Xs)Hs dBs+

∫ t

0

Φ′(Xs)Ks ds+ 12

∫ t

0

Φ′′(Xs)H2s ds .

(VI.4)

The last term is known as the Ito correction, by comparison with theusual calculus (chain rule) formula, which we recover by taking H = 0 .


Note that formally, one can write :

dΦ(Xt) = Φ′(Xs)Hs dBs + Φ′(Xs)Ks ds+ 12 Φ′′(Xs)H

2s ds

= Φ′(Xs) dXs + 12 Φ′′(Xs)H

2s ds .

The formula looks like a second order Taylor formula with “(dBt)2 = dt ”.

For example, for any real a, b and any t ≥ 0 (by linearity, Φ can beC-valued as well) :

e√−1 a (Bt+b t) = 1 +

√−1 a

∫ t

0

e√−1 a (Bs+b s)dBs + (

√−1 ab− a2

2)

∫ t

0

e√−1 a (Bs+b s)ds .

Before proving the Ito formula, let us notice that it can be extendedby an important method called localisation.

Corollary VI.5.5 Ito’s Formula (VI.4) holds in S for any C2 functionΦ and semimartingale X ∈ S such that (Φ′(Xs)Hs), (Φ′(Xs)Ks) and(Φ′′(Xs)H

2s ) are in Λ .

Proof If H,K are bounded, we can define a sequence of stoppingtimes Tn ≤ n , increasing to infinity and such that for 0 ≤ s ≤ Tn : Xs ,Φ′(Xs)Hs , Φ′(Xs)Ks and Φ′′(Xs)H

2s are bounded by n ; then apply

Ito’s Formula in Sb to (Xt∧Tn), and let n go to infinity. Convergencefollows straightforwardly from the Ito isometric identity (VI.3). Thegeneral case is obtained by approaching H,K by bounded processes.

Proof of Theorem VI.5.4 : Using the localisation argument given in theproof of Corollary VI.5.5, we can assume that X is bounded. Sinceon any bounded interval we can find a sequence of polynomials (Φn)such that Φn and its two first derivatives approximate uniformly Φ andits two first derivatives, we see that it is enough to show the formulawhen Φ is a polynomial. Therefore, by induction on the degree of Φ,


it is enough to show that for two processes (Xt) and (X t) in Sb , thefollowing integration by parts formula holds :

XtX t−x0 x0 =

∫ t

0

(XsHs+XsHs)dBs+

∫ t

0

(XsKs+XsKs+HsHs)ds .

Indeed, applying this to X = ΦX completes the induction. Now, usingbilinearity it is enough to consider the case X = X . Finally, writing

X2t − x2

0 = 2n∑k=1

X t(k−1)n

(X tk

n−X t(k−1)

n

)+

n∑k=1

(X tk

n−X t(k−1)

n

)2

,

we see that the result is a direct consequence of Lemma VI.5.3. Example We get : eaBt−

a2

2 t = 1 + a

∫ t

0

eaBs−a2

2 s dBs , for any a ∈ R

and t ≥ 0 . This shows that Brownian exponential martingales arestochastic integrals (added to 1).

In particular, for any stopping time τ , eaBτ∧t−a2

2 τ∧t is a square integrablemartingale. As an application, we can compute the law of the hittingtime Tx of level x by the standard Brownian motion starting at 0. Then

E(eaBTx∧t−

a2

2 Tx∧t)

= 1. By symmetry, take ax > 0, let t → ∞ , and use

dominated convergence, to conclude that E(e−

a2

2 Tx)

= e−|a x|.

Exercise Find the law of the hitting point of a given line by a planarBrownian motion.

The Ito formula can easily be generalized to functions of d variablesand to the case where we consider r independent Brownian motions Bp.The space Sb (respectively S) of semimartingales of bounded (respec-tively L2) type becomes the space of square integrable continuous pro-

cesses which can be written as a sum x0 +r∑p=1

∫ t

0

Hps dB

ps +

∫ t

0

Ks ds ,


with bounded cadlag adapted processes (respectively processes in Λ) Hp

and K. Thus we have the following.

Theorem VI.5.6 (Ito’s Formula) Consider d semimartingales in Sb :

Xjt := xj0 +

r∑p=1

∫ t

0

Hj,ps dBp

s +

∫ t

0

Kjs ds , 1 ≤ j ≤ d ,

and a C2 function Φ with bounded derivatives on Rd. Then Φ(Xt) ≡Φ(X1

t , . . . , Xdt ) belongs to Sb , and we have :

Φ(Xt)− Φ(x0) =d∑j=1

r∑p=1

∫ t

0

Φ′j(Xs)Hj,ps dBp

s +d∑j=1

∫ t

0

Φ′j(Xs)Kjs ds

+ 12

∑1≤i,j≤d

r∑p=1

∫ t

0

Φ′′i,j(Xs)Hi,ps Hj,p

s ds .

(VI.5)

Proof The proof is the same as for Theorem VI.5.4, except that we have to consider also

terms of the form :n∑k=1

(∫ tkn

t(k−1)n

Hps dB

ps

)(∫ tkn

t(k−1)n

Hqs dB

qs

), for 1 ≤ p < q ≤ r , to establish

that they converge to 0 in L2. To prove this, as in the proof of Lemma VI.5.2, we first check

that E[(∫ t

sHpu dB

pu

)(∫ t

sHqu dB

qu

) ∣∣∣∣Fs] = 0 , which is obvious for step processes. Then

we observe that by the Schwarz inequality and Lemma VI.5.1, if Hp and Hq are uniformly

bounded by C, then E[∣∣∣∣∫ t

sHpudB

pu

∣∣∣∣2 × ∣∣∣∣∫ t

sHqu dB

qu

∣∣∣∣2]≤ 9C4 (t− s)2 .

A statement similar to Corollary VI.5.5 holds for S.

Notation If Xt := x0 +r∑p=1

∫ t

0

Hps dB

ps +

∫ t

0

Ks ds is in S, one uses

the notation 〈X,X〉t :=r∑p=1

∫ t

0

|Hps |2 ds .


If X t := x0 +r∑p=1

∫ t

0

Hps dB

ps +

∫ t

0

Ks ds is another element of S , we

use the notation :⟨X,X

⟩t

:=r∑p=1

∫ t

0

Hps H

ps ds .

If J is a cadlag adapted process such that (HpsJs) and (KsJs) belong to

Λ, one uses the notation

∫ t

0

Js dXs :=r∑p=1

∫ t

0

JsHps dB

ps +

∫ t

0

JsKs ds .

In this way, the multidimensional Ito formula in Sb (or in S, when itholds) can be written as :

Φ(Xt)− Φ(x0) =∑i

∫ t

0

Φ′i(Xs) dXis + 1

2

∑i,j

∫ t

0

Φ′′i,j(Xs) d⟨X i, Xj

⟩s.

(VI.6)

Note in particular that the quadratic covariation⟨X,X

⟩t

is bilinear,and note also the following integration by parts formula : for any X inS and Y in Sb , X Y belongs to S, and we have :

Xt Yt = x0 y0 +

∫ t

0

Yt dXt +

∫ t

0

Yt dXt + 〈X, Y 〉t . (VI.7)

Observe that by (VI.7) uniqueness of the decomposition in the space S holds, since if

Xt =

r∑p=1

∫ t

0Hps dB

ps +

∫ t

0Ks ds = 0 , then we get

r∑p=1

∫ t

0

(Hps

)2ds = 〈X,X〉t = 0 and then

r∑p=1

∫ t

0Hps dB

ps =

∫ t

0Ks ds = 0 .

The following application of Ito’s Formula provides a characterizationof d-dimensional Brownian motion.

Theorem VI.5.7 (P. Levy) Let M jt := xj0 +

r∑p=1

∫ t

0

Hj,ps dBp

s ,


1 ≤ j ≤ d , be d martingales in S such that 〈M j,Mk〉t = t 1j=k forall 1 ≤ j, k ≤ d and t ∈ R+ .

Then M 1, . . . ,Md are d independent Brownian motions(in other words,

(M 1, . . . ,Md) is a d-dimensional Brownian motion).

Proof Ito’s Formula (VI.6) (extended as in Corollary VI.5.5) ensures

that Zt := exp

[d∑j=1

λjMjt − 1

2

d∑j,k=1

λjλk〈M j,Mk〉t]

defines a martingale

for any sequence of real numbers λ1, . . . , λd . Fix t0 ∈ R+ and A ∈ Ft0 ,and consider the martingales

N jt := 1A × (M j

t −M jt∧t0) =

∫ t

t∧t01A dM

js ,

which obey : 〈N j, Nk〉t = 1A×(t−t∧t0)×1j=k. Changing in Zt aboveM into N and λ := (λ1, . . . , λd) into

√−1 λ , we get the martingale

Zt = exp

[√−1

d∑j=1

λjNjt + 1

2 |λ|2(t− t ∧ t0) 1A

]

= 1A exp

[√−1

d∑j=1

λj(Mjt −M j

t0) + 12 |λ|2(t− t0)

]+ 1Ac .

Now E(Zt) = 1 reads

E[1A× exp

(√−1

d∑j=1

λj(Mjt −M j

t0))]

= P(A)× exp(− 1

2 |λ|2(t− t0)).

This shows the independence relations and the Gaussian character whichguarantee the claim.

Remark VI.5.8 It is sometimes convenient to extend the Ito integral to an even largerspace. We say that a continuous process (Mt) is a local martingale if and only if thereexists a sequence of stopping time Tn increasing to infinity such that for all n , (Mt∧Tn) is


a martingale of M∞c . It is in fact equivalent to produce a localising sequence such that thestopped processes are martingales, since then we can build from it and from hitting times ofintegers τn by M another localizing sequence Tn ∧ τn satisfying the assumption. Of course,martingales are local martingales. Let Λ0 be the space of previsible processes such that∫ t

0X2s ds <∞ for any t > 0 almost surely. The Ito integral extends to a linear map from

Λ0 into the space of local martingales. Ito’s Formula also extends to this framework, but itis often less directly useful, as the Ito integral in it is then only a local martingale.

We shall need the following Burholder-Davis-Gundy inequality.

Proposition VI.5.9 Fix p ≥ 1 . There exists a positive constant Cpsuch that for any cadlag adapted process H ∈ Λ and for any s, t ≥ 0 ,we have :

E

[∣∣∣∣∫ t

s

H dB

∣∣∣∣2p]≤ Cp E

[ ∣∣∣∣∫ t

s

H2

∣∣∣∣p ] .Proof Up to modifying H, we can take s = 0 , for simplicity. Consider

the continuous martingale Mt :=

∫ t

0

H dB . Note that using a non-

decreasing sequence of stopping times going almost surely to infinity, wecan assume that H and M are bounded. Applying Ito’s Formula (VI.4)to M , we get :

|Mt|2p = 2p

∫ t

0

|Ms|2p−1 sign(Ms)Hs dBs + p(2p− 1)

∫ t

0

|Ms|2(p−1)H2s ds

whence, taking expected value and using Holder’s inequality :

E[|Mt|2p

]= p(2p− 1)E

[∫ t

0

|Ms|2(p−1)H2s ds

]

≤ p(2p− 1)E[

sup0≤s≤t

|Ms|2(p−1)

∫ t

0

H2s ds

]

VI.6. THE STRATONOVITCH INTEGRAL 195

≤ p(2p− 1)E[

sup0≤s≤t

|Ms|2p](p−1)/p

× E[∣∣∣∣∫ t

0

H2

∣∣∣∣p]1/p

.

Using Doob’s inequality (recall Theorem VI.3.2), we obtain :

E[|Mt|2p

]≤ p(2p− 1)E

[(2p

2p−1

)2p

|Mt|2p](p−1)/p

× E[∣∣∣∣∫ t

0

H2

∣∣∣∣p]1/p

whence finally :

E[|Mt|2p

]≤(

(2p)2p−1

2(2p−1)2p−3

)p× E

[∣∣∣∣∫ t

0

H2

∣∣∣∣p] . Proposition VI.5.10 For any d ≥ 2 , almost surely a d-dimensional Brownian motion(Bs) never vanishes at any positive time s .

Proof It is obviously enough to consider a 2-dimensional Brownian motion (Bs) = (B1s , B

2s ).

Suppose first that it starts at B0 6= 0 , and denote by τ its hitting time of 0. Apply-ing Ito’s Formula (VI.6) to log |Bt| = 1

2 log(|B1

t |2 + |B2t |2)

and recalling that log is har-

monic in R2, we get for 0 ≤ t < τ : log |Bt| =∫ t

0

B1s dB

1s +B2

s dB2s

|B1s |2 + |B2

s |2. For any n ∈ Z ,

let τn denote the hitting time of the sphere of radius e−n (centred at 0). If P(τ <∞) > 0 , fix N ∈ N∗ such that P(τ−N > τ) > 0 . Fix also n0 > − log |B0| . Forany n ≥ n0 , (log |Bt∧τ−N∧τn |) is a bounded continuous martingale, and by PropositionVI.4.5, (log |Bτ−N∧τn |)n≥n0 is a discrete martingale. Now since log |Bτ−N∧τn | = N1τ−N<τn−n1τ−N>τn , we have E

[log |Bτ−N∧τn0

|]

= N − (N + n)P(τ−N > τn)−→ −∞ as n→∞ ,

a contradiction which proves that τ =∞ almost surely.

Suppose now that B0 = 0 . By the strong Markov property (Corollary VI.3.5) and the above

applied to every (Bτn+t −Bτn), we see that almost surely (Bs) never vanishes after time τn .

This yields the claim, since by continuity of (Bs) at 0 we have limn→∞

τn = 0 almost surely.

VI.6 The Stratonovitch integral

This modification of the Ito integral proves to be convenient to express the solutions oflinear stochastic differential equations (and for stochastic calculations on manifolds, see [IW],Section VI.6).


Definition VI.6.1 Let B1, . . . , Bd be d independent standard real Brownian motions, andX,Y belong to S. The Stratonovitch integral of X with respect to Y is defined almostsurely , for all t ∈ R+, by : ∫ t

0X dY :=

∫ t

0X dY + 1

2 〈X,Y 〉t . (VI.8)

Proposition VI.6.2 Let B1, . . . , Bd be d independent standard real Brownian motions,

X1, . . . , Xn belong to Sb , X := (X1, . . . , Xn), and F be a function of class C3 on Rn,having bounded derivatives. Then almost surely for all t ∈ R+ we have :

F (Xt) = F (X0) +n∑j=1

∫ t

0

∂F

∂xj(Xs) dXj

s . (VI.9)

Proof Apply Definition (VI.8) to

∫ t

0

∂F

∂xj(Xs) dXj

s , Ito’s Formula (VI.5) to∂F

∂xj(Xs)

and to F (Xs), to get :∫ t

0

∂F

∂xj(Xs) dXj

s =

∫ t

0

∂F

∂xj(Xs) dX

js + 1

2

∫ t

0

∂2F

∂xj∂xk(Xs) d〈Xj , Xk〉s = F (Xs)− F (X0).

Remark VI.6.3 In an analogous way, the so-called backward integral is given by

2

∫ t

0X dY −

∫ t

0X dY . It happens to be the limit in probability of the stochastic Rieman-

niann sumsN−1∑j=0

Xtj+1(Ytj+1−Ytj ), in the spirit of Lemma VI.5.3. Equivalently,

∫ t

0X dY is

the limit in probability of the stochastic Riemanniann sumsN−1∑j=0

12 (Xtj +Xtj+1)(Ytj+1−Ytj ).

VI.7 Notes and comments

The martingale theory was developed by Doob, in connection withpotential theory [Do].

Brownian motion was studied by Einstein and Langevin, in relationwith heat transport and kinetic theory of gas. The construction of the

VI.7. NOTES AND COMMENTS 197

associated measure on continuous paths is due to Wiener. See also Levy[Lev].

There is an extensive literature on Brownian motion. Sample pathsproperties such as Holderianity, the existence of quadratic variation, thestructure of the set of zeros, the existence of local times. . . have been ex-tensively studied and are described in many references. See for example[Lev], [IMK], [Bi], [RY], [KS], [Dt]. Besides, Brownian motion allowsa reinterpretation of potential theoretical notions as capacity, balayage,potentials, etc. See [PSt], [Do].

Stochastic calculus was introduced by Bernstein [Be], Ito [I] and Gih-man [Gi1], [Gi2]. See the first pages of [IW] for a historical account. Ashort classical exposition was given by McKean [MK]. Short introduc-tions are given in [Bi], [Dt].

The theory can be extended to Levy processes (with Poisson inte-grals), Markov processes [KW], and beyond [Me]. A lot of treatises orlectures have been published, some of them including a general study ofstochastic processes. See in particular [GS], [J], [IW], [KS], [RY], [RW2],[Ka], [O], [Pr].

Chapter VII

Brownian motions on groups ofmatrices

The main application of Ito’s calculus is to provide solutions to sto-chastic differential equations driven by Wiener processes (and more gen-erally semi-martingales). Existence and uniqueness is based as in thedeterministic case on Picard’s iteration method. The simplest equationsare the linear ones, and we construct left and right Brownian motionson groups of matrices, as solutions to linear stochastic differential equa-tions. We establish in particular that the solution of such an equationlives in the subgroup associated with the Lie subalgebra generated bythe coefficients of the equation. We also consider reversed processes,Hilbert-Schmidt estimates, approximation by stochastic exponentials,Lyapounov exponents and diffusion processes.

Then we concentrate on important examples : the Heisenberg group,PSL(2), SO(d), PSO(1, d), the affine group Ad and the Poincare groupPd+1. By a projection we then obtain the spherical and hyperbolic Brow-nian motions, and the relativistic diffusion in Minkowski’s space.

The relativistic diffusion is a continuous process on the relativisticphase space. Its law is invariant under any Lorentz transformation.

199

200 CHAPTER VII. BROWNIAN MOTIONS ON GROUPS OF MATRICES

It can be lifted to the Poincare group, where it describes the motionof a solid submitted to infinitely many infinitely small boosts whichare stationary, uniformly distributed and independent, in the properframe and proper time of the solid. (Note that this diffusion should notbe viewed as a model for physical Brownian motion in the relativisticsetting. Indeed, physical Brownian motion is produced by the boostscoming from the interaction with a medium. They should not occurstationarily in the proper time of the solid).

VII.1 Stochastic Differential Equations

We consider Stochastic Differential Equations (S.D.E.’s for short) ofthe homogeneous type :

Xxs = x+

∫ s

0

σ(Xxt ) dBt +

∫ s

0

b(Xxt ) dt ,

where (Bt) is a Brownian Motion (B.M. for short) of Rd′, σ is a functionfrom Rd into L(Rd′,Rd), b is a function from Rd into Rd, x ∈ Rd, s runsR+, and the unknown is the process (Xx

s )s≥0 . This is also denoted by :

dXt = σ(Xt) dBt + b(Xt) dt .

Theorem VII.1.1 Fix two functions σ, b on Rd, σ L(Rd′,Rd)-valued,b Rd-valued, which are globally Lipschitz : there exists a constant Csuch that :∥∥σ(x)− σ(x′)

∥∥+∣∣b(x)− b(x′)

∣∣ ≤ C |x− x′| , for all x, x′ ∈ Rd.

Let x ∈ Rd, and (Bt) be a Brownian motion of Rd′, with filtration(Fs := σBt | 0 ≤ t ≤ s

). Then there exists a unique continuous

VII.1. STOCHASTIC DIFFERENTIAL EQUATIONS 201

(Fs)-adapted process (Xxs ) such that :

Xxs = x+

∫ s

0

σ(Xxt ) dBt +

∫ s

0

b(Xxt ) dt , for any s ≥ 0 .

Proof Let us drop the index x, and fix the Euclidean norm, as normon Rd,Rd′, and on L(Rd′,Rd) ≡ Rdd′. Set X0

s ≡ x , and for n ∈ N :

Xn+1s := x+

∫ s

0

σ(Xnt ) dBt +

∫ s

0

b(Xnt ) dt ,

and

Ens := E

[sup

0≤t≤s

∣∣Xn+1t −Xn

t

∣∣2].In particular, we have :

E0s = E

[sup

0≤t≤s

∣∣σ(x)Bt+b(x)t∣∣2] ≤ 2

∥∥σ(x)∥∥2

d′∑j=1

E[

sup0≤t≤s

|Bjt |2]

+2s2|b(x)|2

≤ 2(‖σ(0)‖+ C|x|

)2(d′ × 4s) + 2s2

(|b(0)|+ C|x|

)2

= O[(1 + |x|)2(4d′ + s)s

]<∞ .

Let us proceed by induction, fixing any n ∈ N∗ and supposing that forsome constant C ′ ≥ 2C2 :

En−1u ≤

(1 + |x|

)2 ×[(4d′ + s)C ′

]n × un/n! , for 0 ≤ u ≤ s .

Thus sup0≤v≤·

∥∥σ(Xnv )− σ(Xn−1

v )∥∥ is square integrable, so that∫ ·

0

[σ(Xn

v )− σ(Xn−1v )

]dBv is a square integrable Rd-valued martingale

(recall Section VI.4 ; we mean that its components in the canonical basisof Rd are square integrable martingales), to which we want to applyDoob’s inequality (Theorem VI.3.2) and Ito’s isometric Identity (recall


Remark VI.4.4). Theorem VI.3.2 was written for real square integrablemartingales, and Remark VI.4.4 dealed with real Brownian motion, buta vector version is easily deduced, as follows. Writing for short[

σ(Xnv )− σ(Xn−1

v )]

=: φ(v) =((φ(v)ij

))1≤i≤d,1≤j≤d′ ,

by Theorem VI.3.2 and Remark VI.4.4 we have indeed :

E

[sup

0≤t≤u

∣∣∣∣ ∫ t

0

[σ(Xn

v )− σ(Xn−1v )

]dBv

∣∣∣∣2]

= E

[sup

0≤t≤u

d∑i=1

∣∣∣∣ d′∑j=1

∫ t

0

φ(v)ij dBjv

∣∣∣∣2]

≤ d′d∑i=1

d′∑j=1

E

[sup

0≤t≤u

∣∣∣∣ ∫ t

0

φ(v)ij dBjv

∣∣∣∣2]≤ 4d′

d∑i=1

d′∑j=1

E[ ∫ u

0

∣∣φ(v)ij∣∣2dv]

= 4d′E[ ∫ u

0

∥∥σ(Xnv )− σ(Xn−1

v )∥∥2dv

].

Using this and the Schwarz inequality, we get for 0 ≤ u ≤ s :

Enu = E

[sup

0≤t≤u

∣∣∣∣ ∫ t

0

[σ(Xn

v )−σ(Xn−1v )

]dBv +

∫ t

0

[b(Xn

v )− b(Xn−1v )

]dv

∣∣∣∣2]

≤ 2E[

sup0≤t≤u

∣∣∣∣ ∫ t

0

[σ(Xn

v )− σ(Xn−1v )

]dBv

∣∣∣∣2]

+ 2E[

sup0≤t≤u

∣∣∣∣ ∫ t

0

[b(Xn

v )− b(Xn−1v )

]dv

∣∣∣∣2]

≤ 8d′E[ ∫ u

0

∥∥σ(Xnv )−σ(Xn−1

v )∥∥2dv

]+ 2E

[u

∫ u

0

∣∣b(Xnv )− b(Xn−1

v )∣∣2 dv]

≤ 2C2 × (4d′ + u)×∫ u

0

E[|Xn

v −Xn−1v |2

]dv ≤ C ′(4d′ + s)

∫ u

0

En−1v dv

≤ (1 + |x|)2 × [(4d′ + s)C ′]n+1 × un+1/(n+ 1)! ,

VII.1. STOCHASTIC DIFFERENTIAL EQUATIONS 203

which proves by induction that for 0 ≤ u ≤ s :

Enu ≤ (1 + |x|)2

[(4d′ + s)C ′

]n+1 un+1

(n+ 1)!.

Hence the series∑n

√Ens converges, the variable

∑n

sup0≤t≤s

|Xn+1t −Xn

t |is integrable, and then the sequence Xn converges almost surely uni-formly on compact subsets of R+ , to a continuous process X, which isnecessarily the wanted solution. In particular, it is clearly (Fs)-adapted.

As to uniqueness, we see in the same way as above, if X, Y are twosolutions, that :

Es := E(

sup0≤t≤s

∣∣Xt − Yt∣∣2) ≤ (4d′ + s)C ′

∫ s

0

Eu du .

Hence, if Es ≤ N is finite, we get Eu ≤ N[(4d′ + s)C ′

]n unn!

(for 0 ≤u ≤ s) by induction, whence Es = 0 for any s > 0 , which means thatX and Y are indistinguishable.

Now we can reduce the proof to this case, by localisation, using thestopping times TN := inf

t∣∣ |Xt|+ |Yt| >

√N

: indeed, we can applythe above to (XN

t , YNt ) := (Xt∧TN , Yt∧TN ), and to

Es(N) := E(

sup0≤t≤s

∣∣XNt − Y N

t

∣∣2) = E(

sup0≤t≤s∧TN

∣∣Xt − Yt∣∣2) ≤ N .

This shows that almost surely Xt = Yt for all t ∈ [0, TN ], and then forall t ≥ 0 since lim

N→∞TN =∞ , by continuity of X, Y .

We prove now a comparison theorem for real S.D.E.’s.

Consider a real (Ft)-Brownian motion (Bt), and Lipschitz real functionsσ , b on R , so that Theorem VII.1.1 applies, guaranteeing existence and


uniqueness of the real diffusion (Xs), strong(i.e., (Ft)-adapted

)solution

to the following stochastic differential equation :

Xs = X0 +

∫ s

0

σ(Xt) dBt +

∫ s

0

b(Xt) dt .

Consider moreover two adapted continuous processes (β1(t)), (β1(t)), andreal (Ft)-adapted processes (X1

s ), (X2s ), solving the following stochastic

differential equations (for j = 1, 2) :

Xjs = Xj

0 +

∫ s

0

σ(Xjt ) dBt +

∫ s

0

βj(t) dt .

Theorem VII.1.2 Suppose the following comparison assumption al-most surely holds :

β1(t) ≤ b(X1t ) and b(X2

t ) ≤ β2(t) for all t ≥ 0 , and X10 ≤ X2

0 .

Then almost surely : X1t ≤ X2

t for all t ≥ 0 .

Proof By the usual localisation argument, we can suppose that σ and b are uniformlybounded and Lipschitz. For any n ∈ N∗, denote by ϕn a continuous function from R into

[0, 2n], null on ]−∞, 0] ∪ [1/n,∞[, and such that

∫ 1/n

0ϕn = 1 . Set then

φn(x) :=

∫ x

0dy

∫ y

0ϕn . Note that (x− 1

n)+ ≤ φn(x) ≤ x+ for any x ∈ R .

Applying Ito’s Formula, we have :

φn(X1s −X2

s ) = φn(X1s −X2

s )− φn(X10 −X2

0 )

=

∫ s

0φ′n(X1

t −X2t )[σ(X1

t )− σ(X2t )]dBt

+

∫ s

0φ′n(X1

t −X2t )[β1(t)− β2(t)

]dt+ 1

2

∫ s

0ϕn(X1

t −X2t )∣∣σ(X1

t )− σ(X2t )∣∣2dt .

The first integral of the right hand side has mean 0, while the third is O(1/n), since :

0 ≤∫ s

0ϕn(X1

t −X2t )∣∣σ(X1

t )− σ(X2t )∣∣2 dt

VII.2. LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS 205

≤∫ s

0(2n)× sup

∣∣σ(x)− σ(x′)∣∣2∣∣∣ |x− x′| ≤ 1/n

.

As to the second one : Js :=

∫ s

0φ′n(X1

t −X2t )[β1(t)− β2(t)

]dt , we have :

Js ≤∫ s

0φ′n(X1

t −X2t )[b(X1

t )− b(X2t )]dt ≤ C

∫ s

01X1

t>X2t |X

1t −X2

t | dt

= C

∫ s

0(X1

t −X2t )+dt .

Hence, letting n→∞, we get :

E[(X1

s −X2s )+]≤ C

∫ s

0E[(X1

t −X2t )+]dt , whence E

[(X1

s −X2s )+]

= 0 for all s ≥ 0 . This

yieds the result.

VII.2 Linear Stochastic Differential Equations

Let us here specialize Theorem VII.1.1 to the case of main interest forthe following : linear equations. The main feature will be that, takingfor such equation convenient coefficients in some Lie subalgebra G, theresulting process (solving the equation) will live in the associated sub-group (even if this is not a Lie subgroup), yielding a so-called Brownianmotion on this group.

Theorem VII.2.1 Consider A0, A1, . . . , Ak ∈ M(d), X0 ∈ M(d),and an Rk-valued standard Brownian motion Wt = (W 1

t , . . . ,Wkt ).

Then there exists a unique continuous M(d)-valued (σWt | 0 ≤ t ≤ s)-adapted process (Xs), solution of :

Xs = 1 +k∑j=1

∫ s

0

XtAj dWjt +

∫ s

0

Xt

(12

k∑j=1

A2j + A0

)dt . (VII.1)

Moreover the right increments of (Xs) are independent and homoge-neous : for any t ≥ 0 , the process s 7→ X−1

t Xs+t has the same law


as the process s 7→ Xs , and is independent of the σ-field Ft generatedby the Brownian motion (Ws) up to time t .

Furthermore this statement remains true if the constant time t is re-placed by a stopping time T , conditionally on the event T <∞.

Proof The first part is merely a particular case of Theorem VII.1.1.Then for any fixed r > 0 we have :

Xr+s = Xr +k∑j=1

∫ r+s

r

XtAj dWjt +

∫ r+s

r

Xt

(12

k∑j=1

A2j + A0

)dt ,

or equivalentlyX−1r Xr+s = 1 +

k∑j=1

∫ s

0

X−1r Xr+tAj d(W j

r+t −W jr ) +

∫ s

0

X−1r Xr+t

(12

k∑j=1

A2j + A0

)dt .

Hence X−1r Xr+t satisfies the same equation (VII.1) as Xs , up to shifting

the Brownian motion (Wt) by time r . This proves the result, since thisshifted Brownian motion is Brownian as well, and is independent of Fr .Finally this proof is valid as well with a stopping time T instead of r,restricting to the event T <∞.

Nota Bene The process(WA

t :=k∑j=1

AjWjt

)is a Brownian motion on

the Lie subalgebra G generated by A0, . . . , Ak . Precisely, this is a con-tinuous process with independent and homogeneous increments, and we

have for any u ∈ G∗ and any t ≥ 0 : E[e√−1 u(WA

t )]

= e−t2

2 α(u,u), where

α :=k∑j=1

Aj⊗Aj is a non-negative bilinear form on G∗, which determines

the law of (WAt ).

VII.2. LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS 207(Indeed, considering a basis (E1, . . . , E`) of G, and its dual basis (E∗1 , . . . , E

∗` ), we obtain

the claimed formula by writing : Aj =∑n

λnjEn , u =∑m

µmE∗m ,

u(WAt ) =

∑j,n

µnλnj Wjt , α(u, u) =

∑j

u(Aj)2 =

∑j

[∑n

µnλnj

]2.

)

The matrix A0 of Equation (VII.1) is known as the drift componentof the process (Xs).

Applying Ito’s Formula, we get the following(in which the right Lie

derivatives LA are defined on M(d), as in Section I.1.4).

Theorem VII.2.2 Consider the solution (Xs) to the linear S.D.E.(VII.1), and a function φ of class C2 on M(d). Then we have :

(i) The process (Xs), unique solution of (VII.1), is almost surely GL(d)-valued. (ii) We have

φ(Xs) = φ(1) +k∑j=1

∫ s

0

LAjφ(Xt) dWjt +

∫ s

0

[12

k∑j=1

(LAj)2 +LA0

]φ(Xt) dt

(VII.2)or equivalently, in Stratonovitch form (recall Section VI.6) :

φ(Xs) = φ(1) +k∑j=1

∫ s

0

LAjφ (Xt) dW jt +

∫ s

0

LA0φ (Xt) dt . (VII.3)

The second-order operator A := 12

k∑j=1

(LAj)2 + LA0is the so-called

infinitesimal generator (or generator for short) of the process (Xs).

Proof Let us apply Ito’s Formula (VI.5). We have :

φ(Xs)−φ(1) =∑

1≤a,b≤d

∫ s

0

∂φ

∂Xab(Xt)dX

abt + 1

2

∑1≤a,b,α,β≤d

∫ s

0

∂2φ

∂Xab ∂Xαβ(Xt) d〈Xab, Xαβ〉t


=∑

1≤a,b≤d

∫ s

0

∂φ

∂Xab(Xt)×

[k∑j=1

(XtAj)ab dW j

t +

(Xt

(12

k∑j=1

A2j+A0

))abdt

]

+ 12


k∑j=1

∫ s

0

∂2φ

∂Xab ∂Xαβ(Xt) (XtAj)

ab (XtAj)αβ dt

=k∑j=1

∫ s

0

LAjφ(Xt) dWjt +

∫ s

0

(12

k∑j=1

LA2j

+ LA0

)φ(Xt) dt

+ 12


k∑j=1

∫ s

0

[(LAj)2φ(Xt)− LA2

jφ(Xt)

]dt ,

since

LAφ(X) =dodεφ(X eεA) =

dodεφ(X + εXA) =

∑1≤a,b≤d

∂φ

∂Xab(X) (XA)ab

and(LA)2φ(X)− LA2φ(X)

=∑a,b

dodε

[∂φ

∂Xab(X + εXA)

((X + εXA)A

)ab]−∑a,b

∂φ

∂Xab(X)(XA2)ab

=∑

1≤a,b,α,β≤d

∂2φ

∂Xab ∂Xαβ(Xt) (XA)ab(XA)αβ.

This proves Formula (VII.2). Formula (VII.3) follows at once from Formula

(VII.2) (applied to φ and to LAjφ) and from Formula (VI.8) (of Definition VI.6.1).

To get the first assertion of the statement, let us apply Formula (VII.2)to the particular function : φ := [X 7→ det(X)]. We clearly have

LA det(X) =dodε

det(1 + εA) det(X) = Tr(A) det(X) ,

so that


det(Xs) = 1 +k∑j=1

∫ s

0

det(Xt)Tr(Aj) dWjt +

∫ s

0

[12

k∑j=1

(Tr(Aj))2 + Tr(A0)

]det(Xt) dt ,

and then

det(Xs) = exp

[k∑j=1

Tr(Aj) dWjs + Tr(A0) s

]almost surely, since by Ito’s Formula the right-hand side satisfies theabove equation for det(Xs), and since by Theorem VII.1.1 the solutionto this equation is unique. This shows that almost surely det(Xs) doesnot vanish.

Note that the process (Xs) solving the linear S.D.E. (VII.1) enjoys also the following

trivial left invariance property : for any g ∈ M(d), (g Xs) is the solution to the equation

(VII.1) where the starting matrix 1 = X0 is replaced by g = X0 .

Theorem VII.2.3 Consider the solution (Xs) to the linear S.D.E.(VII.1), with coefficients A0, A1, . . . , Ak belonging to some Lie subal-gebra G. Then the process (Xs) takes almost surely its values in thegroup associated to G (recall Section I.1.4).

It is called a left Brownian motion on G (with drift A0).

Proof Let V0 ⊂⊂ V ⊂⊂ V ′ be compact neighbourhoods of 0 inM(d)(V0 ⊂⊂ V meaning that V0 is included in the interior of V), such thatthe restriction of the exponential map to a neighbourhood of V ′ be adiffeomorphism. For 0 ≤ t < τV ′ := inf

t > 0

∣∣Xt /∈ exp(V ′)

we haveMt := exp−1(Xt) ∈ V ′, and d expMt

is an isomorphism fromM(d) ontoTexp(Mt)GL(d) ≡ exp(Mt) × M(d), which is given (recall PropositionI.1.3.2) by :

d expMt(B) = exp(Mt)×

∑k∈N

ad(−Mt)k

(k + 1)!(B) =: exp(Mt)× αMt

(B) .


Moreover, the linear map αMtdepends analytically on Mt , so that its

inverse α−1Mt

is well defined for 0 ≤ t < τV ′ and depends analytically onMt as well. For such t we have :

LAj exp−1(Xt) = dodε exp−1(Xt + εXtAj) = d exp−1

Xt(XtAj)

= (d expMt)−1(XtAj) = α−1

Mt(Aj).

Hence, applying Formula (VII.3) of Theorem VII.2.2 with φ = exp−1,we get, for 0 ≤ s < τV ′ :

Ms =k∑j=1

∫ s

0

α−1Mt

(Aj) dW jt +

∫ s

0

α−1Mt

(A0) dt .

Consider now a smooth function ψ from M(d) into [0, 1], equal to1 on V0 and vanishing outside V , and then the stochastic differentialStratonovitch equation :

M ′s =

k∑j=1

∫ s

0

[α−1M ′t

(Aj)×ψ(M ′t)]dW j

t +

∫ s

0

α−1M ′t

(A0)ψ(M ′t) dt , (VII.4)

where M ′t =: x1

t V1 + · · · + x`t V` belongs to G, (V1, . . . , V`) being somefixed basis of G. Note that the matrices α−1

M ′t(Aj) × ψ(M ′

t) are welldefined for any time t , by the choice of the function ψ which let themvanish when M ′

t does not belong to V . Moreover they belong to the Liesubalgebra G, since the restriction of αM ′t to G is an automorphism of G.Furthermore they depend smoothly on M ′

t , which allows to compute theIto correction in the above Stratonovitch integral, according to (VI.8),as follows :

12

k∑j=1

⟨[α−1M ′t

(Aj)× ψ(M ′t)],W j

t

⟩

= 12

k∑j=1

[ψ(M ′

t) d[α−1· (Aj)

]M ′t

+ α−1M ′t

(Aj) dψM ′t

](〈M ′

t ,Wjt 〉)


= 12 ψ(M ′

t)k∑j=1

[ψ(M ′

t) d[α−1· (Aj)

]M ′t

+ α−1M ′t

(Aj) dψM ′t

](α−1M ′t

(Aj)).

Hence Equation (VII.4), seen as an equation relating to (x1t , . . . , x

`t) ∈

R`, satisfies the hypothesis of Theorem VII.1.1. Therefore, applyingTheorem VII.1.1 provides a solution M ′

s which lives in G for all times s(and is constant after the random time inft > 0 |M ′

t /∈ V).

Now by the above, Mt satisfies the same equation on the random timeinterval [0, τV [ , with τV = infs > 0 |Ms /∈ V. Thus, by the unique-ness of the solution to this localised equation, viewing it as an equationrelating to M(d) ≡ Rd2

and applying Theorem VII.1.1, we must haveMt = M ′

t , and then Mt ∈ G and Xt ∈ G, for 0 ≤ t < τV . G denoteshere the group associated to G .

In particular, we have Xt ∈ G for 0 ≤ t ≤ τV0. Since the process

X−1τV0XτV0

+t =: X ′t satisfies the same equation as Xt by Theorem VII.2.1,we can apply the above to it, to get : X ′t ∈ G for 0 ≤ t ≤ τV0

. Thisyields : Xt ∈ G for 0 ≤ t ≤ τV2

0(where V2

0 = MM ′|M,M ′ ∈ V0).By an obvious induction, we thus obtain : Xt ∈ G for 0 ≤ t ≤ τVn0 .The result follows, since the increasing sequence of neighborhoods (Vn0 )exhausts M(d), so that the sequence τVn0 increases to infinity.

Remark VII.2.4 Theorem VII.2.2(and in particular Ito’s Formula

(VII.2))

is valid on C2(G). Note that the above theorem VII.2.3 isvalid even if the group G is not closed.

Remark VII.2.5 The laws (νt) of the left Brownian motion (Xt) con-stitute a convolution semi-group on G : νs+t = νs ∗ νt , and we haveddt νt(f) = νt(Af), for any test-function f on G. The (infinitesimal)generator A (of Theorem VII.2.2) is thus associated with (νt).


Proof The second formula follows at once from Formula (VII.2), andfor any non-negative s, t , using Theorem VII.2.1 we have :∫

G

f d(νs ∗ νt) =

∫G2

f(gh) νs(dg) νt(dh)

=

∫G

E[f(g X−1

s Xs+t)]νs(dg) = E

[f(Xs+t)

]=

∫G

f dνs+t .

Definition VII.2.6 Consider the left Brownian motion (Xs) solvingthe linear S.D.E. (VII.1), and its generator A (recall Theorem VII.2.2).The associated semi-group (Pt)t≥0 is defined by Ptf(g) = E

[f(gXt)

], for

any f ∈ Cb(G) and g ∈ G.

Note that Ptf(g) makes sense for f ∈ Cb(M(d)

)and g ∈M(d) as well.

Owing to Remark VII.2.5, Pt acts by convolution with the law νt ofXt : Ptf = f ∗ νt .Nota Bene From now on, we assume that the group G is closed inM(d) (hence, is a Lie group). Note that this implies that the functionsin C2

b (G) admit a C2b -continuation on some neighbourhood of G.

Proposition VII.2.7

(i) (Pt) is a family of non-negative endomorphisms on Cb(G), such thatP0 is the identity and Pt1 = 1 for any t ≥ 0 .

(ii) (Pt) satisfies the so-called semi-group property : PsPt = Ps+t .

(iii) The semi-group (Pt) is strongly continuous : limt0‖Ptf − f‖ = 0 ,

for any f ∈ Cb(M(d)

).

(iv) For any n ∈ N∗, g ∈ G, f0 , . . . , fn ∈ Cb(G) and 0 ≤ t1 ≤ . . . ≤ tn ,we have :

E[f1(g Xt1)× · · · × fn(g Xtn)

]= Pt1

[f1 Pt2−t1

[f2 . . . Ptn−tn−1

fn]]

(g).


Proof (i) is clear. (ii) follows from the independence and homogeneityof the right increments of (Xs) : for any s, t ≥ 0 we have

PsPtf(g) = E[Ptf(gXs)

]= E

[f(gXsX

−1s Xs+t)

]= Ps+tf(g) ,

which is precisely semi-group property(which is equivalent to that of

the convolution semi-group (νt)).

(iii) By Equation (VII.2), (Pt) satisfies :

Psφ = φ+

∫ s

0

PtAφ dt , for any φ ∈ C2b (M(d)), s ≥ 0 .

This implies the strong continuity of (Pt) by the closedness of G, recallthe Nota Bene above.

(iv) Proceeding as for the semi-group property we have :

E[f1(g Xt1) . . . fn(g Xtn)

]= E

[f1(g Xt1) . . . fn−2(g Xtn−2

)×(fn−1 × (Ptn−tn−1

fn))(g Xtn−1

)].

The claim follows, by an obvious induction on n .

Proposition VII.2.8 The semi-group (Pt) is Fellerian on G , whichmeans that in addition of the properties in the preceding propositionVII.2.7, it maps C2

b (G) into C2b (G). Moreover we have

d

dsPsφ = PsAφ = APsφ for any φ ∈ C2

b (G), s ≥ 0 .

This means in particular that C2b (G) is contained in the so-called domain

of the (infinitesimal) generator A , i.e., the space of those f ∈ Cb(G) forwhich there exists Af ∈ C(G) such that lim

t0

∥∥1t (Ptf −f)−Af

∥∥ = 0(so

that the graph of A is the closure of that of its restriction to C2b (G)

).


Proof The first claim is clear from the very definition VII.2.6 of (Ps),by dominated convergence. Then the first equality follows at once fromEquation (VII.2), as expressed in the proof of (iii) of the precedingproposition VII.2.7. Finally we deduce the second equality of the state-ment again from the very definition of (Ps) by dominated convergence,using the expression or A (in Theorem VII.2.2) and noticing that theprocess (Xt) has moments of order 2, see Lemma VII.2.12 below.

Remark VII.2.9 Equation (VII.1) of Theorem VII.2.1 was taken withincrements on the right. We can consider similarly the analogous equa-tion with increments on the left :

Ys = 1−k∑j=1

∫ s

0

Aj Yt dWjt +

∫ s

0

(12

k∑j=1

A2j − A0

)Yt dt . (VII.5)

Then Theorem VII.2.1 is valid as well, Theorem VII.2.2 and Formula(VII.2) hold with the left Lie derivatives −L′A replacing the right Liederivatives LA , so that the process (Ys) has infinitesimal generator

12

k∑j=1

(L′Aj)2 − L′A0.

Theorem VII.2.1 holds, with the left increments Ys+t Y−1t considered in-

stead of the right increments X−1t Xs+t (and with right invariance instead

of left invariance). Theorem VII.2.3 is valid as well : the process (Ys) isalmost surely G-valued if the coefficients Aj belong to G .

The process (Ys) is then called a right Brownian motion on G (with driftcomponent −A0).

Proposition VII.2.10 The right Brownian motion (Yt) of Equation(VII.5) is almost surely the inverse of the left Brownian motion (Xt) ofEquation (VII.1) : Xt Yt = 1 almost surely for all t ≥ 0 .


Proof Applying Ito’s (integration by parts) Formula to Xt Yt and usingEquations (VII.5) and (VII.1) we obtain :

Xt Yt − 1 =

∫ t

0

Xs dYs +

∫ t

0

dXs Ys + 〈X, Y 〉t

=k∑j=1

∫ t

0

Xs

[− Aj Ys dW

js +

(12

k∑j=1

A2j − A0

)Ys ds

]

+k∑j=1

∫ t

0

[XsAjdW

js +Xs

(12

k∑j=1

A2j+A0

)ds

]Ys−

k∑j=1

∫ t

0

XsA2jYsds = 0.

Proposition VII.2.11 Consider the solution (Xs) to the linear S.D.E.

(VII.1), fix r > 0 , and set Xrs := X−1

r−sXr and Ws := Wr−s −Wr , for

0 ≤ s ≤ r . Then the reversed process(Xrs

)satisfies the following

backward equation : for 0 ≤ s ≤ r ,

Xrs = 1−

k∑j=1

∫ s

0

Aj Xrt dW

jt +

∫ s

0

(12

k∑j=1

A2j + A0

)Xrt dt . (VII.6)

As a consequence, if A0 = 0 , for fixed s > 0 the variable X−1s has

the same law as the variable Xs . More generally, for any A0 and fixeds > 0 , the variable X−1

s has the same law as the solution X∗s of :

X∗s = 1 +k∑j=1

∫ s

0

AjX∗t dW

jt +

∫ s

0

(12

k∑j=1

A2j − A0

)X∗t dt .

Nota Bene The identity in law between Xs and X−1s can generally hold

only for fixed s , since in a non-Abelian group they do not have theirindependent increments on the same side. See a precise counter-examplein the Nota Bene of Section VII.6.3 below.


Proof Note that(Ws

)is also a Brownian motion, with filtration

F[s,t] := σWs′−Ws

∣∣ s ≤ s′ ≤ t

= σWu | r−t ≤ u ≤ r−s = F[r−t,r−s]

so that by Remark VII.2.9, Equation (VII.6) possesses a unique solution.Recall that we have

Xr = Xr−s +k∑j=1

∫ r

r−sXtAj dW

jt +

∫ r

r−sXt

(12

k∑j=1

A2j + A0

)dt ,

whence, setting Zs := (Xr)−1s = X−1

r Xr−s :

Zs = 1−k∑j=1

∫ r

r−sX−1r XtAj dW

jt −

∫ r

r−sX−1r Xt

(12

k∑j=1

A2j + A0

)dt .

Consider now a dyadic regular partition r − s = s′0 < . . . < s′N = r of[r − s, r], and set sn := r − s′N−n , so that 0 = s0 < . . . < sN = s is apartition of [0, s], and approach the Ito stochastic integrals by Riemannsums

(recall Lemma VI.5.3(ii)

)as follows :∫ r

r−sXt dW

jt = lim

N→∞

N−1∑n=0

Xs′n

(W j

s′n+1−W j

s′n

)

= limN→∞

N∑n=1

Xr−sN−n+1

[W j

sN−n− W j

sN−n+1

]

= − limN→∞

N−1∑n=0

Xr−sn+1

[W j

sn+1− W j

sn

]=

− limN→∞

N−1∑n=0

Xr−sn

[W j

sn+1−W j

sn

]− limN→∞

N−1∑n=0

(Xr−sn+1−Xr−sn)

[W j

sn+1−W j

sn

].


Hence ∫ r

r−sX−1r Xt dW

jt =

− limN→∞

N−1∑n=0

Zsn

[W j

sn+1− W j

sn

]− lim

N→∞

N−1∑n=0

(Zsn+1− Zsn)

[W j

sn+1− W j

sn

].

Observe now that[W j

sn+1− W j

sn

]∈ F[r−sn+1,r−sn] , while Xr−sn ∈ Fr−sn.

Since by Theorem VII.2.1, Zsn = X−1r Xr−sn is independent from Fr−sn

which contains F[r−sn+1,r−sn], Lemma VI.5.3(ii) yields

limN→∞

N−1∑n=0

Zsn

[W j

sn+1− W j

sn

]=

∫ s

0

Zt dWjt .

Then by polarization (using bilinearity) Lemma VI.5.3(i) yields

limN→∞

N−1∑n=0

(Zsn+1− Zsn)

[W j

sn+1− W j

sn

]=

∫ s

0

d⟨Z, W j

⟩t.

Thus we get :∫ r

r−sX−1r Xt dW

jt = −

∫ s

0

Zt dWjt −

∫ s

0

d⟨Z, W j

⟩t.

Notice this is just minus the backward integral (of Z·, see Remark VI.6.3).

This shows first that the martingale part of Zs isk∑j=1

∫ s

0

ZtAj dWjt ,

and then that

−∫ r

r−sX−1r Xt dW

jt =

∫ s

0

Zt dWjt +

∫ s

0

ZtAj dt .

Hence we got so far :

Zs = 1+k∑j=1

∫ s

0

ZtAj dWjt +

k∑j=1

∫ s

0

ZtA2j dt−

∫ r

r−sZr−t

(12

k∑j=1

A2j+A0

)dt


= 1 +k∑j=1

∫ s

0

ZtAj dWjt +

∫ s

0

Zr−t

(12

k∑j=1

A2j − A0

)dt .

Now, using Proposition VII.2.10 we see that Xrs = Z−1

s solves Equation(VII.5) with (−A0) instead of A0 , i.e., precisely Equation (VII.6).

Assume finally A0 = 0 , so that Equations (VII.5) and (VII.6) are thesame, up to a Brownian change. Then Xr = Xr

r has the same law asYr = X−1

r (recall Proposition VII.2.10).

We shall need to control the L2-norm of the solution to Equation(VII.1) (and recall that we already used this for proving PropositionVII.2.8). Note that the Hilbert-Schmidt norm we use here on matrices,and denote by ‖ · ‖HS in the following lemma, is merely the Euclidean

norm on M(d) seen as Rd2

:∥∥X∥∥

HS:=√

Tr(tX X) =∥∥tX∥∥

HS.

Lemma VII.2.12 For any x ∈ Rd and A,X ∈M(d) we have :

|Ax| ≤∥∥A∥∥

HS|x| , i.e.,

∥∥A∥∥HS

dominates the Euclidean operator normof A ; and moreover :∥∥AX∥∥

HS≤∥∥A∥∥

HS

∥∥X∥∥HS

,∣∣LA∥∥X∥∥HS∣∣ ≤ ∥∥A∥∥HS ∥∥X∥∥HS ,

and∣∣∣(LA)2

[∥∥X∥∥2

HS

]∣∣∣ ≤ 4∥∥A∥∥2

HS

∥∥X∥∥2

HS.

As a consequence, the process Xt solving Equation (VII.1) in SectionVII.2 satisfies the following L2-norm control : for any t ≥ 0,

E[∥∥Xt

∥∥2

HS

]≤ d× exp

[2(∥∥A0

∥∥HS

+k∑j=1

∥∥Aj

∥∥2

HS

)t

].

Proof A direct application of Schwarz’ inequality in Rd yields : |Ax| ≤ ‖A‖HS |x| , andTr(XY ) ≤ ‖X‖HS ‖Y ‖HS . Note that ‖tXX‖HS ≤ ‖X‖2HS , as is easily seen by writingtXX = P D tP with P tP = 1 and D diagonal and non-negative. This implies that

‖XY ‖2HS = Tr(tY tXXY

)= Tr

(Y tY tXX

)≤ ‖Y tY ‖HS ‖tXX‖HS ≤ ‖X‖2HS ‖Y ‖2HS .


Otherwise LA[‖X‖2HS

]= Tr

[tA tXX + tXX A

]= Tr

[(tA+A) tXX

], and

L2A

[‖X‖2HS

]= Tr

[tA2 tXX + 2 tA tXXA+ tXX A2

]= Tr

[(tA2 + 2A tA+A2) tXX

],

so that by Schwarz’ inequality∣∣∣LA[‖X‖2HS]∣∣∣ ≤ ‖tA+A‖HS × ‖X‖2HS ≤ 2 ‖A‖HS × ‖X‖2HS , and∣∣∣L2A‖X‖2HS

∣∣∣ ≤ ‖tA2 + 2A tA+A2‖HS ‖X‖2HS ≤[‖tA2‖HS + 2 ‖A tA‖HS + ‖A2‖HS

]‖X‖2HS

≤ 4 ‖A‖2HS ‖X‖2HS .

Set K := ‖A0‖HS +k∑j=1‖Aj‖2HS , and for any N ∈ N : TN := inf

t ≥ 0

∣∣ ‖Xt‖2HS ≥ N

.

Then applying Ito’s Formula (VII.2)(with φ(X) = ‖X‖2HS and s = t ∧ TN

), taking expec-

tation(note that the martingale part of ‖Xt∧TN ‖2HS has its L2-norm bounded by 2N

√Kt)

and using the above estimates, we get for any N ∈ N, t ≥ 0 :

E[∥∥Xt∧TN

∥∥2

HS

]≤∥∥1∥∥2

HS+ E

[ ∫ t∧TN

02K

∥∥Xs

∥∥2

HSds

]≤ d+ 2K

∫ t

0E[∥∥Xs∧TN

∥∥2

HS

]ds ,

whence by (Gronwall) classical iteration : E[∥∥Xt∧TN

∥∥2

HS

]≤ d e2Kt, and then by N → ∞

and Fatou’s Lemma : E[‖Xt‖2HS

]≤ d e2Kt.

The following simple observation will be useful in the following section VII.3, and inSection X.2.

Remark VII.2.13 For any M(d)-valued continuous process (At), any real Brownian mo-tion (Wt), and any 0 ≤ u < v , we have :

E[∥∥∥∥∫ v

uAt dWt

∥∥∥∥2

HS

]=

∫ v

uE[∥∥At∥∥2

HS

]dt .

Proof By the very definition of ‖ · ‖HS and by Ito’s formula and linearity of the trace, wehave :

E[∥∥∥∥∫ v

uAt dWt

∥∥∥∥2

HS

]= E

[Tr

(∫ v

u

tAt dWt ×∫ v

uAt dWt

)]

= E[ ∫ v

uTr

(∫ t

u

tA dW×At)dWt +

∫ v

uTr

(tAt ×

∫ t

uA dW

)dWt +

∫ v

uTr(tAtAt)dt

)]=

∫ v

uE[∥∥At∥∥2

HS

]dt .


VII.3 Approximation of left B.M. by exponentials

We consider here the left Brownian motion (Xt), solution to Equation (VII.1), with co-efficients A0, A1, . . . , Ak in a given Lie sub-algebra G. According to Theorem VII.2.1 andTheorem VII.2.3, this is almost surely a G-valued process, with homogeneous and indepen-

dent right increments X−1t Xt+s , and generator A = 1

2

k∑j=1

(LAj )2 − LA0 .

Of course, we could as well have written this section for right Brownian motion (Yt)(solving Equation (VII.5)) instead. The changes are straightforward, as it already was todeduce Remark VII.2.9 from the preceding of Section VII.2.

In the present section we get the left Brownian motion (Xt) as a limit of products ofexponentials. In particular, this provides alternative proofs to Theorem VII.2.3, stating that(Xt), (Yt) belong to the Lie subgroup G (the closed subgroup generated by exp[G]), and tothe fact that for any fixed time t , Xt and X−1

t have the same law (Proposition VII.2.11).

The matrices Aj and the Rk-valued Brownian motion W = (W 1, . . . ,W k) being fixed,for any times t < t′, let us denote the exponential we need by :

EWA (t, t′) := exp

[k∑j=1

Aj(Wjt′ −W

jt ) +A0(t′ − t)

]. (VII.7)

Lemma VII.3.1 The norm ‖ · ‖HS being as in Lemma VII.2.12, we have :

E[∥∥∥EWA (t, t′)

∥∥∥2

HS

]≤ 2 exp

[2

(∥∥A0

∥∥HS

+

k∑j=1

∥∥Aj∥∥2

HS

)(t′ − t)

].

Proof By Lemma VII.2.12, we have for any A ∈M(d) :∥∥ exp(A)∥∥HS≤∑n

∥∥∥∥Ann!

∥∥∥∥HS

≤∑n

‖A‖nHSn!

= exp[‖A‖HS

],

so that(using E

[e2aWt′−t

]= e2a2(t′−t)) :

E[∥∥∥EWA (t, t′)

∥∥∥2

HS

]≤ E

exp

[2

∥∥∥∥ k∑j=1

Aj(Wjt′ −W

jt ) +A0(t′ − t)

∥∥∥∥HS

]≤ E

[exp

[2

k∑j=1

∥∥Aj∥∥HS |W jt′ −W

jt |+ 2

∥∥A0

∥∥HS

(t′ − t)]]

VII.3. APPROXIMATION OF LEFT B.M. BY EXPONENTIALS 221

≤ 2 exp

[2(∥∥A0

∥∥HS

+k∑j=1

∥∥Aj∥∥2

HS

)(t′ − t)

].

We can develop the exponential of Formula (VII.7) up to second order, as follows.

Lemma VII.3.2 The norm ‖ · ‖HS being as in Lemma VII.2.12, we have for any t, ε > 0 :

EWA (t, t+ ε) = 1 +k∑j=1

AjWj,tε + εA0 +

ε

2

k∑j=1

A2j + 1

2

k∑i,j=1

(AiAj +AjAi)

∫ ε

0W i,ts dW j,t

s +Rεt ,

where W j,ts := (W j

t+s −W jt ), and E

[∥∥Rεt∥∥2

HS

]= O(ε3) (uniformly with respect to t).

In particular, we have E[∥∥∥EWA (t, t+ ε)− 1

∥∥∥2

HS

]= O(ε2) (uniformly with respect to t).

Proof By definition (VII.7) of EWA , we have :

EWA (t, t+ ε) = 1 +k∑j=1

AjWj,tε +A0 ε+ 1

2

k∑i,j=1

AiAjWi,tε W j,t

ε +Rεt ,

with

Rεt :=ε

2

k∑j=1

(A0Aj +AjA0)W j,tε +

ε2

2A2

0 +∞∑n=3

1

n!

[ k∑j=1

AjWj,tε +A0 ε

]n.

By Ito Formula, we have

W i,tε W j,t

ε =

∫ ε

0W i,ts dW j,t

s +

∫ ε

0W j,ts dW i,t

s + ε 1i=j ,

so thatk∑

i,j=1

AiAjWi,tε W j,t

ε =k∑

i,j=1

(AiAj +AjAi)

∫ ε

0W i,ts dW j,t

s + εk∑j=1

A2j .

Hence we have to control the terms of the remainder Rεt . By Lemma VII.2.12, we have first :

E

∥∥∥∥ε k∑j=1

(A0Aj +AjA0)W j,tε

∥∥∥∥2

HS

≤ 2k ε2 ‖A0‖2HSk∑j=1

‖Aj‖2HS E[|W j,t

ε |2]

= O(ε3).

Then,

E

∥∥∥∥[ k∑j=1

AjWj,tε +A0 ε

]n∥∥∥∥2

HS

1/2

≤ E

∥∥∥∥ k∑j=1

AjWj,tε +A0 ε

∥∥∥∥2n

HS

1/2


≤ E[( k∑

j=1

∥∥Aj∥∥HS |W j,tε |+ ε

∥∥A0

∥∥HS

)2n]1/2

≤ E[[

(k + 1)n−1

( k∑j=1

∥∥Aj∥∥nHS |W j,tε |n + εn

∥∥A0

∥∥nHS

)]2]1/2

≤ (k + 1)n−1

( k∑j=1

∥∥Aj∥∥nHS√E[|W j,t

ε |2n]

+ εn∥∥A0

∥∥nHS

)

≤[(k + 1)

√ε]n( k∑

j=1

2n/2∥∥Aj∥∥nHS √n! + εn/2

∥∥A0

∥∥nHS

),

since E[|W j,t

ε |2n]

= εn ×n∏j=1

(2j − 1) ≤ (2ε)n × n! . Setting K :=√

2 max0≤j≤k

∥∥Aj∥∥HS ,

and using that√E[‖ · ‖2HS

]is a norm, this yields finally :

E

∥∥∥∥ ∞∑n=3

1

n!

[ k∑j=1

AjWj,tε +A0 ε

]n∥∥∥∥2

HS

1/2

≤∞∑n=3

[(k + 1)K

√ε]n k√n! + εn/2

n!= O(ε3/2).

The last assertion follows at once.

Let τn := 0 = t0 < t1 < . . . < tn < tn+1 = s be a subdivision of a given time interval[0, s], and consider the GL(d)-valued process (Znt ) inductively defined by Zn0 = 1 , and :

Znt = Zntq × EWA (tq, t) for 0 ≤ q ≤ n and tq ≤ t ≤ tq+1 .

Lemma VII.3.3 Set B := 12

k∑j=1

A2j +A0. We have in L2-norm, for any u ∈ [0, s] :

Znu − 1−k∑j=1

∫ u

0Znt Aj dW

jt −

∫ u

0Znt B dt = Mn

u + eCsO(u√|τn|

),

where |τn| := max|tq+1 − tq|

∣∣ 0 ≤ q ≤ n denotes the mesh of the subdivision τn ,

Mnu := 1

2

n∑q=0

Zntq ×k∑

i,j=1

(AiAj +AjAi)

∫ u∧tq+1

u∧tq(W i

t −W itq) dW

jt

is a M(d)-valued centred continuous martingale, and C :=∥∥A0

∥∥HS

+

k∑j=1

∥∥Aj∥∥2

HS.


Proof (After [MK]) Let us fix ` ∈ 0, . . . , n such that t` ≤ u ≤ t`+1 , and set

Nnu :=

n∑q=0

Zntq ×[EWA (u∧ tq, u∧ tq+1)−1−

k∑j=1

Aj (W ju∧tq+1

−W ju∧tq)−B (u∧ tq+1−u ∧ tq)

].

We have :

Λnu := Znu − 1−k∑j=1

∫ u

0Znt Aj dW

jt −

∫ u

0Znt B dt−Nn

u

= Znt` − 1−k∑j=1

∫ t`

0Znt Aj dW

jt −

∫ t`

0Znt B dt−Nn

t`

−Znt`[ k∑j=1

∫ u

t`

(EWA (t`, t)− 1)Aj dWjt −

∫ u

t`

(EWA (t`, t)− 1)B dt

]

=

`−1∑q=0

Zntq

[−

k∑j=1

∫ tq+1

tq

(EWA (tq, t)− 1)Aj dWjt −

∫ tq+1

tq

(EWA (tq, t)− 1)B dt

]

+Znt`

[−

k∑j=1

∫ u

t`

(EWA (t`, t)− 1)Aj dWjt −

∫ u

t`

(EWA (t`, t)− 1)B dt

](by induction)

= −∑q=0

Zntq

[ k∑j=1

∫ u∧tq+1

tq

(EWA (tq, t)− 1)Aj dWjt +

∫ u∧tq+1

tq


].

Hence, setting uq := u ∧ tq (for 0 ≤ q ≤ n+ 1) and

Quq :=k∑j=1

∫ uq+1

uq

(EWA (tq, t)− 1)Aj dWjt +

∫ uq+1

uq

(EWA (tq, t)− 1)B dt ,

and using Lemmas VII.2.12 and VII.3.1, we deduce (by independence of Ftq and Quq ) :√E[∥∥Λnu

∥∥2

HS

]≤

n∑q=0

√E[∥∥Zntq∥∥2

HS

∥∥Quq∥∥2

HS

]=

n∑q=0

√E[∥∥Zntq∥∥2

HS

]×√E[∥∥Quq∥∥2

HS

]

≤√

2 eCs ×n∑q=0

√E[∥∥Quq∥∥2

HS

], with C =

∥∥A0

∥∥HS

+k∑j=1

∥∥Aj∥∥2

HS.


Now, by Remark VII.2.13 we have :

√E[‖Quq‖2HS

]

≤k∑j=1

√E[∥∥∥∥∫ uq+1

uq

(EWA (tq, t)− 1)Aj dWjt

∥∥∥∥2

HS

]+√E[∥∥∥∥∫ uq+1

uq


∥∥∥∥2

HS

]

≤k∑j=1

√E[ ∫ uq+1

uq

∥∥∥(EWA (uq, t)− 1)Aj

∥∥∥2

HSdt

]+√E[ ∫ uq+1

uq

∥∥∥(EWA (uq, t)− 1)B∥∥∥2

HSdt

]

≤k∑j=1

√E[ ∫ uq+1

uq

∥∥EWA (uq, t)− 1∥∥2

HS

∥∥Aj∥∥2

HSdt

]+√E[ ∫ uq+1

uq

∥∥EWA (uq, t)− 1∥∥2

HS

∥∥B∥∥2

HSdt

]

≤( k∑j=1

∥∥Aj∥∥HS +∥∥B∥∥

HS

)×√E[ ∫ uq+1

uq

∥∥EWA (uq, t)− 1∥∥2

HSdt

]

=

[ ∫ uq+1

uq

O(t− uq)2 dt

]1/2

= O(

(uq+1 − uq)3/2),

by Lemma VII.3.2. Hence, so far we have got :√E[∥∥Λnu

∥∥2

HS

]≤√

2 eCsn∑q=0

O(

(uq+1 − uq)3/2)

= eCs × u×O(√|τn|

).

Finally, by Lemma VII.3.2 again, in the same way we have :

Nnu −Mn

u =n∑q=0

Zntq Ruq+1−uquq =

n∑q=0

Zntq ×O(

(uq+1 − uq)3/2)

=√

2 eCsn∑q=0

O(

(uq+1 − uq)3/2)

= eCsO(u√|τn|

),

and the martingale property for (Mnu ) follows at once from the independence between Zntq

and the centred martingale

[tq, tq+1] 3 u 7−→k∑

i,j=1

(AiAj +AjAi)

∫ u∧tq+1

u∧tqW it dW

jt .


Corollary VII.3.4 Notation being as in Lemma VII.3.3, in L2-norm, uniformly with re-spect to u ∈ [0, s], we have :

Znu − 1−k∑j=1

∫ u

0Znt Aj dW

jt −

∫ u

0Znt B dt = eCs ×O

(√u (1 + u) |τn|

).

Proof (After [MK]) By Lemma VII.3.3, we have only to control the centred martingale(Mn

u ). Since(∥∥Mn

u

∥∥HS

)is a non-negative submartingale, we apply Doob’s inequality (The-

orem VI.3.2), to get : for any s′ ∈ [0, s],

E[

max0≤u≤s′

∥∥Mnu

∥∥2

HS

]≤ 4E

[∥∥Mns′∥∥2

HS

]=

∑0≤q,q′≤n

E[Tr(Zntq A

qs′tAq′s′ × tZntq′

)]

where Aqu :=k∑

i,j=1

(AiAj +AjAi)

∫ u∧tq+1

u∧tq(W i

t −W itq) dW

jt . By independence of Aq′s′ from Aqs′ ,

Zntq , Zntq′

as soon as q′ > q , we then have :

E[

max0≤u≤s′

∥∥Mnu

∥∥2

HS

]≤

n∑q=0

E[∥∥Zntq Aqs′∥∥2

HS

]≤

n∑q=0

E[∥∥Zntq∥∥2

HS

∥∥Aqs′∥∥2

HS

]

=n∑q=0

E[∥∥Zntq∥∥2

HS

]E[∥∥Aqs′∥∥2

HS

]≤ 2 e2Cs

n∑q=0

k2k∑

i,j=1

∥∥AiAj+AjAi∥∥2

HS

∫ s′∧tq+1

s′∧tqE[|W i

t−tq |2]dt

≤ k2 e2Csk∑

i,j=1

∥∥AiAj +AjAi∥∥2

HS× s′|τn|

by Lemma VII.3.1 and Remark VII.2.13. This yields the claim.

Theorem VII.3.5 The solution (Xt) to the stochastic linear differential equation (VII.1)is approached in L2-norm by exponentials as follows. For any time S ≥ 0 , we have :

Xs = lim|τn|→0

n∏q=0

exp

[k∑j=1

Aj(Wjtq+1−W j

tq) +A0(tq+1 − tq)], (VII.8)

where |τn| := max|tq+1 − tq|

∣∣ 0 ≤ q ≤ n

denotes the mesh of a generic subdivision τn =0 = t0 < t1 < . . . < tn < tn+1 = s of [0, s], uniformly with respect to s ∈ [0, S].


Proof Using Corollary VII.3.4 and Equation (VII.1), for n ∈ N∗ and s ∈ [0, S] we have :

Zns −Xs =k∑j=1

∫ s

0(Znt −Xt)Aj dW

jt +

∫ s

0(Znt −Xt)B dt+O

(√s |τn|

).

Hence, setting αns := E[

sup0≤u≤s

∥∥Znu −Xu

∥∥2

HS

], we have :

√αns ≤

k∑j=1

√E[

sup0≤u≤s

∥∥∥∥∫ u

0(Znt −Xt)Aj dW

jt

∥∥∥∥2

HS

]+√E[

sup0≤u≤s

∥∥∥∥∫ u

0(Znt −Xt)B dt

∥∥∥∥2

HS

]+O

[√s|τn|

]

≤ 2

k∑j=1

√E[∥∥∥∥∫ s

0(Znt −Xt)Aj dW

jt

∥∥∥∥2

HS

]+√E[[∫ s

0‖(Znt −Xt)B‖HS dt

]2]

+O[√

s|τn|]

by Doob’s inequality (Theorem VI.3.2), and since by Lemma VII.2.12 (for 0 ≤ u ≤ s) :∥∥∥∥∫ u

0ZtB dt

∥∥∥∥2

HS

= Tr

[∫ u

0

∫ u

0

t(ZtB) (Zt′B) dt dt′]

=

∫ u

0

∫ u

0Tr[t(ZtB) (Zt′B)

]dt dt′

≤∫ u

0

∫ u

0

∥∥ZtB∥∥HS ∥∥Zt′B∥∥HS dt dt′ = [∫ u

0

∥∥ZtB∥∥HS dt]2

≤[∫ s

0

∥∥ZtB∥∥HS dt]2

.

Hence, using Remark VII.2.13 we obtain :

√αns ≤ 2

k∑j=1

√E[∫ s

0

∥∥(Znt −Xt)Aj∥∥2

HSdt

]+√E[s

∫ s

0

∥∥(Znt −Xt)B∥∥2

HSdt

]+O

[√s|τn|

]

≤ 2k∑j=1

[∫ s

0αnt∥∥Aj∥∥2

HSdt

]1/2

+

[s

∫ s

0αnt∥∥B∥∥2

HSdt

]1/2

+O[√

s |τn|]

≤[2

k∑j=1

∥∥Aj∥∥HS +√S∥∥B∥∥

HS

]×[∫ s

0αnt dt

]1/2

+O[√

s |τn|].

Setting C ′ := 2

[2

k∑j=1

∥∥Aj∥∥HS +√S∥∥B∥∥

HS

]2

, we therefore have : for any s ∈ [0, S],

αns ≤ C ′∫ s

0αnt dt+ bn s , with bn = O

(|τn|).

Hence fn(s) := e−C′s

∫ s

0αnt dt satisfies f ′n(s) ≤ bn s e−C

′s, whence

VII.4. LYAPOUNOV EXPONENTS 227

fn(s) ≤∫ s

0bn t e

−C′tdt = bn

(1− e−C′s − C ′s e−C′s

)/C ′2 , and then :

E[

sup0≤s≤S

∥∥Zns −Xs

∥∥2

HS

]= αnS ≤ C ′eC

′Sfn(S) + bn S ≤ bn(eC′S − 1)/C ′ = O

(|τn|).

This means as wanted that limn→∞

Zns = Xs in L2-norm, uniformly with respect to s ∈ [0, S].

Remark VII.3.6 It is not much more difficult to show that the convergence of TheoremVII.3.5 occurs also almost surely

(uniformly with respect to s ∈ [0, S]

). See [MK].

VII.4 Lyapounov exponents

Proposition VII.4.1 To the solution (Xs) of the linear S.D.E. (VII.1) (with X0 ∈ GL(d)instead of 1) is associated a so-called Lyapounov exponent, which is a deterministic real λ1

such that

lims→∞

s−1 log∥∥Xs

∥∥ = λ1 ,

almost surely and in L2-norm. Here∥∥ · ∥∥ denotes any norm on M(d). Moreover, we have

|λ1| ≤ (3/2)

k∑j=1

∥∥Aj∥∥2

HS+∥∥A0

∥∥HS

(where the norm ‖ · ‖HS is as in Lemma VII.2.12).

Nota Bene 1) Note from the last formula giving det(Xs) in the proof of Theorem VII.2.2,that we have at once lim

s→∞s−1 log |det(Xs)| = Tr(A0), almost surely and in L2-norm.

2) For λ1 > 0 , the solution is said to be stable ; and unstable for λ1 < 0 .

Proof Let us apply Ito’s Formula (VII.2) to the function log∥∥·∥∥2

HS. Note that by Theorem

VII.2.2(i) and since X0 ∈ GL(d),∥∥X−1

0 Xs

∥∥HS

never vanishes. We obtain :

log∥∥X−1

0 Xs

∥∥2

HS= log d+ 2

k∑j=1

∫ s

0

LAj∥∥X−1

0 Xt

∥∥HS∥∥X−1

0 Xt

∥∥HS

dW jt +

∫ s

0(f1 − f2)(X−1

0 Xt) dt

with f1 :=∥∥ · ∥∥−2

HS×[

12

k∑j=1

(LAj )2 + LA0

](∥∥ · ∥∥2

HS

)and f2 :=

k∑j=1

(LAj∥∥ · ∥∥HS∥∥ · ∥∥HS

)2

.


By Lemma VII.2.12, f1 − f2 is C∞ bounded, since

|f1| ≤ 2

k∑j=1

∥∥Aj∥∥2

HS+ 2

∥∥A0

∥∥HS

and 0 ≤ f2 ≤k∑j=1

∥∥Aj∥∥2

HS,

and the quadratic variation of the martingale part Mt of log∥∥X−1

0 Xs

∥∥2

HSis :

〈M,M〉t = 4k∑j=1

∫ s

0

∣∣∣∣∣LAj∥∥X−1

0 Xt

∥∥HS∥∥X−1

0 Xt

∥∥HS

∣∣∣∣∣2

dt ≤ 4k∑j=1

∥∥Aj∥∥2

HSs =: C s .

As a consequence, on the one hand, for any ε ∈ ]0, 12 [ and for large s , we almost surely have :

s−1 log∥∥X−1

0 Xs

∥∥HS

= O(sε−1/2) + s−1

∫ s

0(f1 − f2)(X−1

0 Xt) dt .

On the other hand, setting for any λ ≥ 1 : Tλ := infs ≥ 1

∣∣M2s > λs2

≥ 1 , and using

Doob’s inequality (Theorem VI.3.2(ii) and Proposition VI.5.9, we obtain :

P[

sups≥1|Ms/s|2 > λ

]= P[Tλ <∞] ≤ λ−2 E

[T−4λ M4

Tλ1Tλ<∞

]≤ λ−2

∑n≥1

n−4 E[M4Tλ

1n≤Tλ<n+1

]≤ (4/3)4 λ−2

∑n≥1

n−4 E[M4n+1

]≤ (4/3)4 λ−2C2

∑n≥1

n−4 E[〈M,M〉2n+1

]≤ (4/3)4 λ−2C2C

2∑n≥1

n−4(n+ 1)2 = C ′ λ−2 .

Hence :∥∥∥ sups≥1|Ms/s|

∥∥∥2

2=

∫ ∞0

P[sups≥1|Ms/s|2 > λ

]dλ ≤ 1 +

∫ ∞1

C ′λ−2 dλ = 1 + C ′ <∞ ,

which proves that sups≥1

∣∣∣s−1 log∥∥X−1

0 Xs

∥∥HS

∣∣∣ is square integrable.

For s > 0, set ϕs := log∥∥X−1

0 Xs

∥∥2

HS, and observe that ϕr+s = ϕr + ϕs Θr − log d ,

whence the sub-additivity property :

ϕr+s ≤ ϕr + ϕs Θr , Θ denoting the shift operator on Brownian trajectories.

As Θ1 preserves the underlying probability and (by the 0 − 1 law) is ergodic, Kingman’ssubadditive ergodic Theorem

(see for example [N2] or [St]

)ensures the almost sure conver-

gence of ϕn/n , to the deterministic ` := infE(ϕn/n)

∣∣n ∈ N∗

, as the integer n goes to

VII.5. DIFFUSION PROCESSES 229

infinity. By the above, this implies |`| ≤ ‖f1−f2‖∞ and the almost sure convergence of ϕs/s(using ϕs

s = [s]s

ϕ[s]

[s] + 1s [ϕs−[s] Θ[s]− log d] and that the random variables sup

0≤h≤1ϕh Θn are

independent identically distributed and integrable)

, as the real s goes to infinity, and then

its L2-convergence too, by dominated convergence.

The result follows, since we have s−1 log ‖Xs‖ = O(s−1) + 12 ϕs/s .

Remark VII.4.2 Using the exterior algebra, it can be proved that Proposition VII.4.1admits the following generalisation, which yields the Lyapounov exponents λ2, . . . , λd afterλ1 : λ1 + · · ·+ λj := lim

s→∞s−1 log ‖X∧js ‖ exists in R, almost surely and in L2-norm, and

is deterministic. For j = d, X∧ds = det(Xs), and then, owing to the Nota Bene followingProposition VII.4.1, we have λ1 + · · ·+ λd = Tr(A0).

VII.5 Diffusion processes

We introduce here an important notion of processes living on a spacewhich does not need to be a group, and consider the particular caseaddressed in this book, of diffusion processes which can be derived froma group-valued Brownian motion by means of some projection.

Definition VII.5.1 (i) A Fellerian (Markovian) semi-group on a sep-arable metric space E is a family (Pt)t≥0 of non-negative continuousendomorphisms on Cb(E), such that : P0 is the identity, Ps+t = Ps Pt ,Pt1 = 1 for any s, t ≥ 0 , and lim

t0‖Ptf − f‖ = 0 for any f ∈ Cb(E).

It extends automatically to the space of non-negative measurable func-tions on E , and the associated kernel is defined for any x ∈ E andmeasurable subset A of E by : Pt(x,A) := Pt1A(x).

(ii) Given a separable metric space E endowed with a Borelian proba-bility measure µ and a Fellerian (Markovian) semi-group (Pt), a con-tinuous E-valued process (Xt) such that

E[f0(X0)×f1(Xt1)×. . .×fn(Xtn)

]=

∫f0 Pt1

[f1 Pt2−t1

[f2 . . .Ptn−tn−1

fn]]dµ


for any n ∈ N∗, f0 , . . . , fn ∈ Cb(E) and 0 ≤ t1 ≤ . . . ≤ tn , is called adiffusion process on E with semi-group (Pt) and initial law µ .

Note that (taking a Dirac mass δx as µ , n=1, and f0 ≡ 1) we have inparticular Ptf(x) = Ex

[f(Xt)

]= E

[f(Xt)

∣∣X0 = x], the index x in

Ex specifying the initial value X0 = x ∈ E .

By Proposition VII.2.7, any left Brownian motion is an example of dif-fusion process

(on a subgroup G of M(d)

).

The other diffusion processes considered in this book are all constructedin the following way (up to a possible left quotient by some Kleiniangroup, as in Sections VIII.1 and VIII.2).

Proposition VII.5.2 Given a left Brownian motion (gt) on a groupG and an independent random variable g on G with law ν, consider acontinuous map p : G → E such that for any g ∈ G the law Pt(x, ·)of p(ggt) depends only on x := p(g) and t . Then (Pt) is a Fellerian(Markovian) semi-group and p(g gt) is a diffusion process on E withsemi-group (Pt) and initial law ν p−1.

One says that the diffusion process starts at x ∈ E when its initial lawis the Dirac mass at x .

Proof We have to prove the identity of Definition VII.5.1(ii), withhere Xt = p(g gt), µ = ν p−1. Setting hk := fk p , we know thatE[hk(g gt)

]= (Ptfk) p(g). Now for any n ≥ 2 and g ∈ G, by indepen-

dence and homogeneity of the right increments g′tn−tn−1:= (gtn−1

)−1gtnwe have :

E[h1(g gt1) . . . hn(g gtn)

]= E

[h1(g gt1) . . . hn−1(g gtn−1

)hn(g gtn−1g′tn−tn−1

)]

= E[h1(g gt1) . . . hn−1(g gtn−1

)× (Ptn−tn−1fn) p(g gtn−1

)]

= E[h1(g gt1) . . . hn−2(g gtn−2

)×(fn−1 × (Ptn−tn−1

fn)) p(g gtn−1

)].

VII.6. EXAMPLES OF GROUP-VALUED BROWNIAN MOTIONS 231

By an obvious induction on n , this yields :

E[h1(g gt1) . . . hn(g gtn)

]= Pt1

[f1 Pt2−t1

[f2 . . .Ptn−tn−1

fn]] p(g).

Therefore by independence of g we obtain :

E[h0(g)h1(g gt1) . . . hn(g gtn)

]=

∫h0 Pt1

[f1 Pt2−t1

[f2 . . .Ptn−tn−1

fn]] p dν

=

∫f0 Pt1

[f1 Pt2−t1

[f2 . . . Ptn−tn−1

fn]]dµ ,

which is the wanted identity of Definition VII.5.1(ii). Note that thesemi-group property of (Pt) follows directly, merely taking n = 2 , f0 =f1 ≡ 1 , and a Dirac mass as µ .

Remark VII.5.3 Note that by Propositon VII.5.2, if (Qt) denotes thesemi-group of (gt) (recall Definition VII.2.6), then for any t ≥ 0 andf ∈ Cb(E) we have (Ptf) p = Qt(f p) .

VII.6 Examples of group-valued Brownian motions

We specialize here the contents of Section VII.2 to some classicalexamples of Lie group, for which a simple expression of some left orright Brownian motion of interest can be given.

VII.6.1 Exponential semimartingale

Take any commuting A0 , A1, . . . , Ak ∈ M(d). Then the associatedleft Brownian motion

(i.e., the corresponding solution (Xs) to the linear

S.D.E. (VII.1))

can be written Xs = exp(A1W1s + · · ·+AkW

ks +A0 s),

as is straightforward from Ito’s Formula (VI.4).


VII.6.2 Left Brownian motion on the Heisenberg group H3

Take k = 2 , A0 = 0 , and A1 :=

0 1 00 0 00 0 0

, A2 :=

0 0 00 0 10 0 0

, so

that [A1, A2] =

0 0 10 0 00 0 0

commutes with A1, A2 , and the Lie algebra

G generated by A1, A2 has basis(A1, A2, [A1, A2]

). Then G = exp(G)

is the group H3 of upper triangular matrices having diagonal (1, 1, 1).Precisely, we have :

[a1, a2, a3] := exp(a1A1 + a2A2 + a3[A1, A2]

)=

1 a1 a3 + a1a2

2

0 1 a2

0 0 1

,

and then the law of H3 is given by :

[a1, a2, a3] ∗ [a′1, a′2, a′3] =

[a1 + a′1 , a2 + a′2 , a3 + a′3 + 1

2(a1a′2 − a′1a2)

],

which is equivalent to the (Campbell-Hausdorff) formula :

exp(A) exp(A′) = exp(A+ A′ + 1

2 [A,A′]), holding for any A,A′ ∈ G .

The associated left Brownian motion(i.e., the solution (Xs) to the

corresponding linear S.D.E. (VII.1))

can be written

Xs =[W 1

s , W2s ,

12

∫ s

0

(W 1

t dW2t −W 2

t dW1t

)],

as is directly verified, dXs = A1 dW1s + A2 dW

2s + [A1, A2]W

1s dW

2s

implying (VII.1). Moreover, setting

Xs,t =[W 1

t −W 1s , W

2t −W 2

s ,12

∫ t

s

((W 1

u −W 1s ) dW 2

u − (W 2u −W 1

s ) dW 1u

)]


for any 0 ≤ s ≤ t , we have Xs = X0,s , (Xs,t)t≥slaw≡ (Xt−s)t≥s for any

s ≥ 0 , and then the easy so-called Chen formula : Xs ∗Xs,t = Xt .

The third component of Xs is the so-called Levy’s area generated bythe planar Brownian motion (W 1

s ,W2s ) ≡ (W 1

s +√−1 W 2

s ).

Theorem VII.6.2.1 Consider a planar Brownian motion B = W +√−1 W ′, started at

0. The Fourier transform of its Levy area At := 12

∫ t

0

[Ws dW

′s −W ′s dWs

]is given by :

E[e√−1 xAt

]=(E[e−(x2/8)

∫ t0 (Ws)2 ds

])2=

1

ch (x t/2).

Proof Consider the planar Brownian motion β +√−1 β′, given by :

βt :=

∫ t

0

Ws dWs +W ′s dW′s

|Bs|; β′t :=

∫ t

0

Ws dW′s −W ′s dWs

|Bs|.

We have d|Bt| = dβt + dt2|Bt| , d arg(Bt) =

dβ′t2|Bt| , and At := 1

2

∫ t

0|Bs| dβ′s .

By Theorem VII.1.1 and since the planar Brownian motion almost surely does not vanishat any positive time (recall Proposition VI.5.10), the first equation can be solved betweentimes t′ and t , as soon as 0 < t′ < t , and then by continuity, between times 0 and t , for anyt > 0 . This shows that

(|Bt|

)is adapted to the filtration generated by (βs), and hence, is

independent of the Brownian motion β′. As a consequence, we get at once :

E[e√−1 xAt

]= E

[e−(x2/8)

∫ t0 |Bs|

2 ds]

= E[e−(x2/8)

∫ t0 W

2s ds−(x2/8)

∫ t0 W

′s2 ds],

which gives the first equality of the statement, by independence of W and W ′.

Note that, by scaling, we clearly have E[e√−1 xAt

]= E

[e√−1 xtA1

].

Now for any real x consider the function : R+ 3 t 7→ φ(t) := ch (xt)− (thx) sh (xt).

We have φ′′ = x2φ , φ > 0 , φ(0) = 1 , φ(1) = 1chx , φ′(1) = 0 . In particular, φ is

convex and decreasing. Hence, ϕ := 1[0,1[ φ + φ(1)1[1,∞[ = maxφ, φ(1) is convex, of classC1, non-increasing. Set F := ϕ′/ϕ . Note that −x shx ≤ F ≤ 0 and F = 0 on [1,∞[ .

We have then :

F (s)W 2s − logϕ(s)− 2

∫ s

0F (t)Wt dWt = F (s)W 2

s −∫ s

0F (t) d(W 2

t )


=

∫ s

0W 2t F′(t) dt =

∫ mins,1

0W 2t F′(t) dt = x2

∫ mins,1

0W 2t dt−

∫ s

0W 2t F

2(t) dt ,

so that the bounded process (Zs) defined by

Zs := exp

[12

(F (s)W 2

s − logϕ(s)− x2

∫ mins,1

0W 2t dt

)]can also be written :

Zs = exp

[ ∫ s

0F (t)Wt dWt − 1

2

∫ s

0F 2(t)W 2

t dt

].

This shows that (Zs) is a local martingale (recall Remark VI.5.8) : indeed, setting

Tn := inft ≥ 0

∣∣ |Wt| = n

, (Zmins,Tn) is a martingale (this is easily seen with Ito’sFormula), so that E(Zmin1,Tn) = 1 . Letting then n go to infinity, since (Zs) is bounded,we get by dominated convergence :

1 = E(Z1) = E(

exp

[12

(F (1)W 2

1 − logϕ(1)− x2

∫ 1

0W 2t dt

)])

= ϕ(1)−1/2 E(

exp

[− x2

2

∫ 1

0W 2t dt

]),

i.e.,

E(

exp

[− x2

2

∫ 1

0W 2t dt

])=√φ(1) =

1√chx

,

which directly gives the claim.

The content of the present section VII.6.2 is easily extended to thecase of the general Heisenberg group H2d+1 ; see for example [EFLJ4],Section 4.4.

VII.6.3 Brownian motions in SL(2)

Take k = 2 , and A1 :=

(12 00 −1

2

), A2 :=

(0 10 0

)in sl(2), so

that [A1, A2] = A2 , and the Lie algebra generated by A1, A2 has basis(A1, A2). Take A0 = aA1 , for a ∈ R.


Then the associated left Brownian motion(i.e., the corresponding solu-

tion (Xs) to the linear S.D.E. (VII.1))

can be written :

Xs =

(√ys xs/

√ys

0 1/√ys

), with ys := exp(W 1

s +as) and xs :=

∫ s

0

yt dW2t .

Similarly, the right Brownian motion solving the linear S.D.E. :

Ys = 1 +

∫ s

0

dWAt Yt +

∫ s

0

(12 (A2

1 + A22) + A0

)Yt dt

can be written :

Ys =

(√ys x′s

√ys

0 1/√ys

), with x′s :=

∫ s

0

y−1t dW 2

t ,

as is easily verified by applying Ito’s Formula. While (X−1s ) solves the

same S.D.E. as (Ys), but with −Aj instead of Aj.

Nota Bene Observe that we have here a precise counter-example to the identity in law of(Xs) and (X−1

s ) jointly for different values of s (even for a = 0, recall the Nota Bene followingProposition VII.2.11). Indeed, if (for 0 < u < s) (Xu, Xs) and (X−1

u , X−1s ) had the same

law, then((X·)

1,1(X·)1,0)u,s

= (xu , xs) and((X−1· )1,1(X−1

· )1,0)u,s

= −(y−1u xu , y

−1s xs)

would have the same law too. Now this is false, since the former is a martingale while thelatter is not, as we compute easily :

E[y−1s xs | Fu

]= E

[e−W

1s

∫ s

0eW

1t dW 2

t

∣∣∣∣Fu]= E

[e−W

1s +W 1

u

∣∣∣Fu] e−W 1u

∫ u

0eW

1t dW 2

t +E[ ∫ s

ueW

1t −W 1

s dW 2t

∣∣∣∣Fu]= e(s−u)/2 y−1

u xu + 0 6= y−1u xu almost surely .

Note that both Brownian motions (Xs) and (Ys) live in the group

T′2 :=T ′x+√−1 y

:= exp(xA2) exp(log y A1)∣∣x ∈ R, y ∈ R∗+

,

which is isomorphic to the affine group A2 (recall Proposition I.4.3).

Since we can never have Xs = −Xs′ or Ys = −Ys′ , these two Brownian motions can be

identified with their images under the canonical projection : SL(2)→ PSL(2), and thus seen

as Brownian motions on PSL(2), or on PSO(1, 2) according to Proposition I.1.5.1.


VII.6.4 Left Brownian motion on SO(d)

Recall from Sections I.2 and I.3 that we see SO(d) as the subgroupof elements in PSO(1, d) which fix e0 , and SO(d− 1) as a subgroup ofof elements in SO(d) which fix e1 .

The solution (Xt) of the linear S.D.E. (for independent standard realBrownian motions W k`) :

Xt = 1 +

∫ t

0

Xs

∑1≤k<`≤d

Ek` dWk`s + 1

2

∫ t

0

Xs

∑1≤k<`≤d

E2k` ds (VII.9)

is a left Brownian motion on SO(d), with infinitesimal generator 12 Ξ0 ,

the half Casimir operator on SO(d). Note that∑

1≤k<`≤dE2k` equals (1−d)

× the orthogonal projector on Rd

(since E2

k`ej = 〈ej, ek〉ek + 〈ej, e`〉e`

for k < ` , and then∑

1≤k<`≤dE2k` ej = (d− 1)

∑1≤k≤d

〈ej, ek〉ek)

.

Proposition VII.6.4.1 Identifying the above SO(d)-valued left Brow-nian motion (Xt) with (σt, %t) ∈ Sd−1 × SO(d − 1) by means of the de-composition Xt = Rσt %t of Proposition III.3.2, we obtain : on the onehand another SO(d)-valued left Brownian motion (Rσt), solving

Rσt = 1 +

∫ t

0

Rσs

d∑k=2

E1k dWks + 1

2

∫ t

0

Rσs

d∑k=2

E21k ds ,

where dW kt := −

d∑=2

〈%t e`, ek〉 dW 1`t (for 2 ≤ k ≤ d) defines a (d − 1)-

dimensional Brownian motion W(rotated under %−1

t from (W 1`))

; on


the other hand, the SO(d− 1)-valued left Brownian motion (%t) solving

%t = 1 +

∫ t

0

%s∑

2≤k<`≤dEk` dW

k`s + 1

2

∫ t

0

%s∑

2≤k<`≤dE2k` ds .

As above,∑

2≤k<`≤dE2k` is (2− d)× the orthogonal projector on Rd−1.

Note that for d ≥ 3 the rotations % mapping e1 to −e1 happen to bepolar : almost surely, we have Xt e1 6= −e1 for all t > 0 ; see RemarkVII.7.2.5 below.

Proof 1) From Equation (VII.9) we get directly :

σt = Rσte1 = Xt e1 = e1 −∫ t

0

Xs

d∑`=2

e` dW1`s + 1

2

∫ t

0

Xs(d− 1)e1 ds

= e1 −∫ t

0

Rσs

d∑`=2

% e` dW1`s + d−1

2

∫ t

0

σs ds .

Hence :

Rσt(e1) = σt = e1 −∫ t

0

Rσs

d∑k=2

ek dWks + d−1

2

∫ t

0

σs ds

=

(1 +

∫ t

0

Rσs

d∑k=2

E1k dWks + 1

2

∫ t

0

Rσs

d∑k=2

E21k ds

)e1 .

As the planar rotation Rσt is prescribed by the image vector σt =Rσt(e1), we obtain

Rσt = 1 +

∫ t

0

Rσs

d∑k=2

E1k dWks + 1

2

∫ t

0

Rσs

d∑k=2

E21k ds .


2) On the other hand, by Proposition VII.2.10, (R−1σt

) solves thefollowing equation (VII.5) :

R−1σt

= 1−∫ t

0

d∑k=2

E1kR−1σsdW k

s + 12

∫ t

0

d∑k=2

E21kR−1

σsds .

Thus by Ito’s integration by parts Formula (VI.7), we have :

%t = R−1σtXt = 1 +

∫ t

0

d(R−1σs

)Xs +

∫ t

0

R−1σsdXs + 〈R−1

σ·, X·〉t

= 1−∫ t

0

d∑k=2

E1k %s dWks +

∫ t

0

%s∑

1≤k<`≤dEk` dW

k`s + 1

2

∫ t

0

d∑k=2

E21k %s ds

+12

∫ t

0

%s∑

1≤k<`≤dE2k` ds−

∫ t

0

d∑j=2

E1j %s∑

1≤k<`≤dEk` d〈W j,W k`〉s .

Now %sE1` %−1s = −

d∑j=2

E1j 〈%s e`, ej〉 , and %s commutes withd∑=2

E1` ,

so that

%t = 1 +

∫ t

0

%s∑

2≤k<`≤dEk` dW

k`s +

∫ t

0

%s

d∑k=2

E21k ds+ 1

2

∫ t

0

%s∑

2≤k<`≤dE2k` ds

−∫ t

0

%s

d∑i,j,m=2

E1i 〈%−1s ej, ei〉

∑1≤k<`≤d

Ek` 〈%sem, ej〉 d〈W 1m,W k`〉s

= 1 +

∫ t

0

%s∑

2≤k<`≤dEk` dW

k`s +

∫ t

0

%s

d∑k=2

E21k ds+ 1

2

∫ t

0

%s∑

2≤k<`≤dE2k` ds

−∫ t

0

%s

d∑i,j,`=2

E1i 〈%−1s ej, ei〉E1` 〈e`, %−1

s ej〉 ds


= 1 +

∫ t

0

%s∑

2≤k<`≤dEk` dW

k`s + 1

2

∫ t

0

%s∑

2≤k<`≤dE2k` ds .

VII.6.5 Left Brownian motions on PSO(1, d) and on Ad

The solution (Λs) of the linear S.D.E. driven by (Ws) ≡ (W 1s , . . . ,W

ds ):

Λs = 1 +

∫ s

0

Λt

d∑j=1

Ej dWjt + 1

2

∫ s

0

Λt

d∑j=1

E2j dt (VII.10)

is a left Brownian motion on the Lorentz-Mobius Lie group PSO(1, d),

which (by Theorem VII.2.2) has generator 12 Ξ := 1

2

d∑j=1

(LEj)2 .

Recall thatd∑j=1

Ej dWjt ≡ dWE

t is a Brownian motion on the Lie algebra

so(1, d) (according to the Nota Bene page 206).

Consider also the solution (Zs) of the linear S.D.E. :

Zs = 1+

∫ s

0

Zt

d∑j=1

Ej dWjt + 1

2

∫ s

0

Zt

( d∑j=1

E2j −(d−1)E1

)dt , (VII.11)

which is a left Brownian motion on the affine group Ad. Recall thatfor convenience we set E1 = E1 . According to Formula (III.3), theinfinitesimal generator of (Zs) and of its associated convolution semi-group νt is

12

[ d∑j=2

(LEj)2 + (LE1

)2 + (1− d)LE1

]= 1

2 D .


Let us apply the Iwasawa decomposition to the left Brownian mo-tion (Λs) solving Equation (VII.10) : we have Λs = Tzs %s , with Tzs =Iw(Λs) ∈ Ad and %s = Iw(Λs) ∈ SO(d).

Theorem VII.6.5.1 (i) The first Iwasawa projection Tzs = Iw(Λs) ofthe left Brownian motion (Λs) is a left Brownian motion on the affinegroup Ad, which solves Equation (VII.11)

(but with the following change

of Brownian motion : the d-dimensional white noise dWt is replaced byits image d(%−1W )t under the random rotation %−1

t

). In particular, the

Hd-valued processes (Λse0) and (Zse0) have the same law. Moreover theRd−1× R∗+-valued process zs = (xs, ys) is given by (for 2 ≤ j ≤ d) :

ys = exp[(%−1W )1

s − d−12 s], and xjs =

∫ s

0

yt d(%−1W )jt . (VII.12)

(ii) The second Iwasawa projection %s = Iw(Λs) of the left Brownianmotion (Λs) is a right Brownian motion on SO(d), the inverse of whichis a left Brownian motion having the same law as the component Rσt

of Proposition VII.6.4.1 (and then can be identified with the sphericalBrownian motion of Proposition VII.6.6.2).

Proof (i) By Equation (VII.10) and Lemma I.4.2 we have :

Tzs e0 = Λs e0 = e0 +

∫ s

0

Tzt

d∑j=1

Ad(%t)[Ej dW

jt + 1

2 E2j dt]%t e0

= e0 +

∫ s

0

Tzt

d∑j=1

[E%tej dW

jt + 1

2 (E%tej)2 dt]e0 .

Now since %t ∈ SO(d) on the one hand we have

d∑j=1

(E%t(ej))2 =

d∑j=1

d∑i,k=1

〈%tej, ei〉〈%tej, ek〉EiEk =d∑

k=1

E2k ,


and on the other hand setting d(%−1W )kt := −d∑j=1

〈%tej, ek〉 dW jt , we find :

d∑j=1

E%tej dWjt = −

d∑j,k=1

〈%tej, ek〉Ek dWjt =

d∑k=1

Ek d(%−1W )kt .

Whence

Tzs e0 = e0 +

∫ s

0

Tzt

d∑j=1

[Ej d(%−1W )jt + 1

2 E2j dt]e0 .

Note that by Levy’s characterization (Theorem VI.5.7) (%−1W ) is an-other d-dimensional Brownian motion, since for 1 ≤ k, ` ≤ d we have :

⟨(%−1W )k, (%−1W )`

⟩t

=

∫ t

0

d∑j=1

〈%sej, ek〉〈%sej, e`〉ds

= −∫ t

0

〈%−1s ek, %

−1s e`〉ds = 1k=` t .

Now using (I.15) : Ej = Ej − E1j (so that Ej e0 = Ej e0) we have :

Tzs e0 − e0 =

∫ s

0

Tzt

d∑j=1

[Ej d(%−1W )jt + 1

2 E2j dt]e0

=

∫ s

0

Tzt d(%−1W )Et e0 + 12

∫ s

0

Tzt dtd∑j=1

E2j e0 .

Then using the commutation relations (I.13) (and E11 = 0) we obtain :

d∑j=1

(E2j − E2

j ) e0 =d∑j=2

((Ej + Ej)(Ej − Ej) + [Ej, Ej]

)e0


=d∑j=2

[Ej, E1j] e0 = −d∑j=2

E1 e0 = −(d− 1)E1 e0 .

So far we have obtained :

Tzs e0 − e0 =

∫ s

0

Tzt

[d∑j=1

Ej d(%−1W )jt + 12

(d∑j=1

E2j − (d− 1)E1

)dt

]e0 .

As by (VII.11) this same equation holds with (Zs) instead of (Tzs) (andW instead of %−1W ), and since both processes are Ad-valued, the firstclaim will follow directly from the uniqueness of the Rd−1× R∗+-valuedprocess (zs) solving such an equation.

To show this uniqueness (which is a priori not obvious on account of theright action on e0), we observe that (since Eje0 = ej and E2

j e0 = e0 ,and noting W instead of %−1W ) this equation reads as well :

Tzs e0 − e0 =

∫ s

0

Tzt

[d∑j=1

ej dWjt + d

2 e0 dt

].

Then Formula (I.19) yields :

ys = 〈Tzse0, e0 + e1〉−1 and xjs = −〈Tzse0, ej〉 ys for 2 ≤ j ≤ d ,

while by Definition I.4.1 :

Tzsej = xjs(e0+e1)+ej and Tzse1 =(ys+

|xs|22 ys

)(e0+e1)− 1

2ys(e0−e1)− xs

ys.

Hence we obtain :

dy−1s = −y−1

s dW 1s + d

2 y−1s ds and d

(xjsys

)= −xjs

ysdW 1

s + dW js + d

2xjsysds ,

whence by Ito’s Formula :

dys = ys dW1s + 2−d

2 ys ds and dxjs = ys dWjs .


These are linear S.D.E.’s, which directly yield the wanted uniquenessand (VII.12) as well.

(ii) By Proposition VII.2.10, (T−1zs

) solves the following equation (VII.5):

T−1zs

= 1−d∑j=1

∫ s

0

Ej T−1ztd(%−1W )jt +

∫ s

0

12

( d∑j=1

E2j + (d− 1)E1

)T−1ztdt .

Thus by Ito’s integration by parts Formula (VI.7), we have :

%s = T−1zs

Λs = 1 +

∫ s

0

d(T−1zt

)Λt +

∫ s

0

T−1ztdΛt + 〈T−1

z·,Λ·〉s

= 1−d∑j=1

∫ s

0

Ej %t d(%−1W )jt +d∑j=1

∫ s

0

%tEj dWjt + 1

2

∫ s

0

%t

d∑j=1

E2j dt

+

∫ s

0

12

( d∑j=1

E2j + (d− 1)E1

)%t dt−

∫ s

0

d∑j,k=1

Ej %tEk d⟨(%−1W )j,W k

⟩t.

Now as in (i) above we have

d∑j=1

∫ s

0

%tEj dWjt =

d∑j=1

∫ s

0

E%tej %t dWjt =

d∑j=1

∫ s

0

Ej %t d(%−1W )jt

and %t

d∑j=1

E2j =

d∑j=1

E2%tej

%t =d∑j=1

E2j %t . Moreover

d∑j,k=1

Ej %tEk 〈(%−1W )j,W k〉t = −d∑

j,k=1

Ej E%tek %t 〈%tek, ej〉 dt =d∑j=1

Ej Ej %t dt .

Therefore(again since

d∑j=1

(E2j − E2

j ) =d∑j=2

(Ej + Ej)E1j + (d− 1)E1

)%s = 1−

d∑j=1

∫ s

0

E1j %t d(%−1W )jt +

∫ s

0

12

[ d∑j=1

(E2j + E2

j − 2EjEj) + (d− 1)E1

]%t dt


= 1−∫ s

0

d∑j=1

E1j %t d(%−1W )jt +

∫ s

0

12

d∑j=1

E21j %t dt .

This means that (%s) is a right Brownian motion on SO(d). Finally byProposition VII.2.10 its inverse (%−1

s ) is a left Brownian motion, whichhas the same law as the component Rσt of Proposition VII.6.4.1.

VII.6.6 Spherical Brownian motion

We exhibit here a first example of diffusion process (which does notlive in a group) in the sense of Section VII.5. We consider the projectionp : % 7→ % e1 from SO(d) onto Sd−1 and the left Brownian motion (Xt)on SO(d) (recall Section VII.6.4). The following shows that they satisfythe assumption of Proposition VII.5.2.

Lemma VII.6.6.1 The law of the process (%Xt e1) depends on % ∈SO(d) only through % e1 .

Actually it is not much more difficult to prove that the law of the SO(d)-valued left Brownian motion (Xt) solving Equation (VII.9) is invariantunder any conjugation by a fixed % ∈ SO(d− 1).

Proof We have to show that the processes (%Xt e1) and (% %Xt e1) havethe same law, for any % ∈ SO(d− 1). Now using Proposition VII.6.4.1,we can replace (Xt) by (%t := Rσt) solving

%t = 1 +

∫ t

0

%s

d∑k=2

E1k dWks + 1

2

∫ t

0

%s

d∑k=2

E21k ds ,

and we then have to deal with %t := % %t %−1 , which solves

%t = 1 +

∫ t

0

%s

d∑k=2

Ad(%)(E1k) dWks + 1

2

∫ t

0

%s

d∑k=2

Ad(%)(E1k)2 ds .


Now on the one hand we have

d∑k=2

Ad(%)(E1k) dWks = −

d∑j,k=2

E1j 〈% ek, ej〉 dW ks =

d∑j=2

E1j d(%−1W

)js,

where %−1W denotes another Brownian motion of Rd−1, rotated fromW under %−1 ; while on the other hand we have

d∑k=2

Ad(%)(E1k)2 =

d∑i,j,k=2

E1iE1j 〈% ek, ei〉〈% ek, ej〉 =d∑

k=2

E21k .

Hence changing (%t) into (%t) merely amounts to rotate its driving (d−1)-

dimensional Brownian motion W by %−1 , which of course does notchange its law, thereby yielding the wanted irrelevance of % .

Lemma VII.6.6.1 allows to apply Proposition VII.5.2, yielding thespherical semi-group, say (St), which satisfies the identity :

Stf(p(%)

)= Qt(f p)(%) = E

[f(%Xt e1)

], (VII.13)

for any t ≥ 0 , % ∈ SO(d) and f ∈ C(Sd−1). As in Remark VII.5.3 (upto the notation), (Qt) denotes here the semi-group of (Xt).

The associated diffusion process is called spherical Brownian motion.When it starts at e1, this is the process (σt) of Proposition VII.6.4.1.Moreover, we have the following analogue of Proposition VII.2.8.

Proposition VII.6.6.2 The spherical Brownian semi-group (St) mapsC2(Sd−1) into C2(Sd−1), and for any f ∈ C2(Sd−1) we have :

d

dtStf = 1

2 ∆Sd−1

Stf = 12 St ∆

Sd−1

f .

Moreover (St) is self-adjoint with respect to the volume measure of Sd−1,and covariant with SO(d) : we have St(f %) = (Stf)% for any t ≥ 0 ,% ∈ SO(d).


By analogy with Proposition VII.2.8, we say that the half sphericalLaplacian 1

2 ∆Sd−1

is the (infinitesimal) generator of the spherical Brow-nian motion and of its semi-group (St).

Proof Note first that a function φ on Sd−1 belongs to C2(Sd−1) if andonly if φ p ∈ C2

(SO(d)

)(since any directional derivative on Sd−1 is

got by using some planar rotation). As by Proposition VII.2.8, Qt mapsC2(SO(d)

)into C2

(SO(d)

), this shows the first claim.

Then by Definition III.4.1 and (VII.13), we have :

d

dt(Stf) p =

d

dtQt(f p) = 1

2 Ξ0Qt(f p) = 12 Ξ0

((Stf) p

)= 1

2 ∆Sd−1

(Stf) p .

In the same way we obtain

d

dt(Stf) p = 1

2 Qt Ξ0(f p) = 12 Qt

(∆Sd−1

f p)

= 12 (St ∆

Sd−1

f) p .

This proves the equalities of the statement. Then by (VII.13) for anyf ∈ C

(SO(d)

)and % , %′ ∈ SO(d) we have :

St(f %)(p(%′)

)= E

[f(%%′Xt e1)

]= Stf

(p(%%′)

)= (Stf) %

(p(%′)

).

Finally the self-adjointness of (St) follows from that of ∆Sd−1

(recallProposition III.6.2.3) : indeed, if (S∗t ) denotes the adjoint semi-group,it is clear thatd

dtS∗t f = 1

2 S∗t ∆Sd−1

f for any f ∈ C2(Sd−1), whence for any 0 ≤ t ≤ s :

d

dtS∗t Ss−tf = 0 , and then S∗sf = Ssf .

VII.6.7 Hyperbolic Brownian motion

We exhibit here a second example of diffusion process, which is es-sential to our purpose in this book. We proceed in a way very similar to


what we did in the preceding section VII.6.6 for the spherical Brownianmotion.

We consider now the projection p = π0 : g 7→ g e0 from PSO(1, d)onto Hd and the left Brownian motion (Λt) on PSO(1, d), of SectionVII.6.5 and Equation (VII.10). The following shows that they satisfythe assumption of Proposition VII.5.2.

Lemma VII.6.7.1 The law of the processes (gΛt e0) and (g Zt e0) de-pend on g ∈ PSO(d) only through g e0 .

Proof Theorem VII.6.5.1(i) allows to substitute (gΛt e0) for (g Zt e0).We have to show the irrelevance of % ∈ SO(d) in the law of %Λt e0 =Λ%t e0 , where Λ%

t := %Λt %−1. Now by the equation (VII.10) of (Λt), we

have

Λ%t = 1 +

∫ t

0

Λ%s

d∑j=1

Ad(%)(Ej) dWjs + 1

2

∫ t

0

Λ%s

d∑j=1

Ad(%)(Ej)2 ds ,

and (as in the proof of Proposition VII.6.6.2) on the one hand we have

d∑j=1

Ad(%)(Ej) dWjs = −

d∑j,k=1

Ek 〈% ej, ek〉 dW js =

d∑k=1

Ek d(%−1W

)ks,

where %−1W denotes another Brownian motion of Rd, rotated from Wunder %−1 ; while on the other hand we have

d∑j=1

Ad(%)(Ej)2 =

d∑i,j,k=1

EiEk 〈% ej, ei〉〈% ej, ek〉 =d∑

k=1

E2k .

Hence changing Λt into Λ%t merely amounts to rotate its driving d-

dimensional Brownian motion W by %−1 , which of course does notchange its law, thereby yielding the wanted irrelevance of % .


Lemma VII.6.7.1 allows to apply Proposition VII.5.2, yielding thehyperbolic heat semi-group, say (Qt), which satisfies the identities :

Qtf(z) = E[f(gΛt e0)

]= E

[f(g Zt e0)

], (VII.14)

for any t ≥ 0, f ∈ Cb(Hd), z ∈ Hd and g ∈ PSO(1, d) such that g e0 = z .

The associated diffusion process is called hyperbolic Brownian motion.We have the following analogue of Propositions VII.2.8 and VII.6.6.2.

Theorem VII.6.7.2 The hyperbolic heat semi-group (Qt) maps C2b (Hd)

into C2b (Hd), and for any f ∈ C2

b (Hd) we have :

d

dtQtf = 1

2 ∆Qtf = 12 Qt ∆f . (VII.15)

Moreover Qt is self-adjoint with respect to the volume measure of Hd,and covariant with PSO(1, d) : we have Qt(f g) = (Qtf) g for anyt ≥ 0 , g ∈ PSO(1, d).

By analogy with Proposition VII.2.8, we say that the half hyperbolicLaplacian 1

2 ∆ is the (infinitesimal) generator of the hyperbolic Brown-ian motion and of its (hyperbolic heat) semi-group (Qt).

Proof Note first that a function φ on Hd belongs to C2b (Hd) if and

only if φ p ∈ C2b

(PSO(1, d)

)(since any directional derivative on Hd

is got by using some planar hyperbolic rotation). As by PropositionVII.2.8, the semi-group Pt maps C2

b

(PSO(1, d)

)into C2

b

(PSO(1, d)

),

this shows the first claim.

Then by Theorem III.5.2, Corollary III.5.3 and (VII.14), we have :

d

dt(Qtf) p =

d

dtPt(f p) = 1

2 ΞPt(f p) = 12 Ξ((Qtf) p

)= 1

2 ∆(Qtf) p .


In the same way we obtain

d

dt(Qtf) p = 1

2 Pt Ξ(f p) = 12 Pt(∆f p

)= 1

2 (Qt ∆f) p .

This proves (VII.15). Then by (VII.14) for any f ∈ Cb(PSO(1, d)

)and

g , g′ ∈ PSO(1, d) we have :

Qt(f g)(p(g′)

)= E

[f(g g′Λt e0)

]= Qtf

(p(gg′)

)= (Qtf) g

(p(g′)

).

Finally the self-adjointness follows from that of ∆ (recall PropositionIII.6.2.3) : indeed, if (Q∗t ) denotes the adjoint semi-group, it is clear thatd

dtQ∗t f = 1

2 Q∗t ∆ f for any f ∈ C2

b (Hd), whence for any 0 ≤ t ≤ s :

d

dtQ∗t Qs−tf = 0 , and then Q∗sf = Qsf .

We describe now the main feature of the asymptotic behaviour ofthe hyperbolic Brownian motion. Recall that the Poincare coordinates(xs, ys) of the hyperbolic Brownian motion (Zs e0) are given by the inte-grated formulas (VII.12).

Proposition VII.6.7.3 For any starting point p ∈ Hd, the hyperbolicBrownian motion (zs) converges almost surely, as s→∞ , to a boundarypoint z∞ ∈ ∂Hd, the law of which is the harmonic measure µp .

Proof First, the covariance with respect to g ∈ PSO(1, d) allows toconsider only the case p = e0 . Then it is clear from Formulas (VII.12)(in which we may and do drop the random rotation %−1

t ) that ys goesalmost surely to 0, and that we almost surely have

xjs −→ xj∞ :=

∫ ∞0

eW1t −d−1

2 t dW jt , for 2 ≤ j ≤ d .

Let us justify this last claim in detail, and beware that xj∞ /∈ L2 (ford = 2 or 3). Note however that Proposition VI.3.3 ensures the almost


sure convergence of

∫ ∞0

e2W 1t −(d−1)t dt . Otherwise, by the Ito isometric

identity, for 0 ≤ s ≤ s′ we almost surely have

E[∣∣xjs′ − xjs∣∣2∣∣∣F1

∞

]=

∫ s′

s

e2W 1t −(d−1)t dt .

Therefore, conditionally on the σ-field F1∞ , (xjs) is almost surely a con-

tinuous martingale(with respect to the filtration (F j

s ))

which convergesto xj∞ in L2

(P(· | F1

∞)), hence almost surely as well by Remark VI.3.8(i).

Finally this provides the wanted almost sure convergence.

The above yields a limit point z∞ = (x∞, 0) ∈ ∂Hd. It remains to com-pute its law. Now by Theorem VII.6.7.2, for any % ∈ SO(d), (% zs) hasthe same law as (zs), so that the the law of z∞ must be SO(d)-invariant.As quoted in Remark III.6.1.3, this yields actually the definition III.6.1.1of µe0

. We postpone to Section X.3 a statement specifying how near from the

limiting geodesic (containing p = z0 and ending at z∞) the Brownianpath is asymptotically located.

VII.7 Relativistic Diffusion

We present now the relativistic diffusion of Minkowski’s space, firstconstructed and studied by Dudley in [D1], [D2] and [D3]. It will appearhere as a projection of some left Brownian motion on the Poincare group,in the same vein as we proceeded with the hyperbolic Brownian motionin the preceding section (letting it appear as a projection of some leftBrownian motion on the Lorentz-Mobius or alternatively, on the affinegroup). We shall then derive its striking structure, of being an integratedhyperbolic Brownian motion, and also the easy part of its asymptoticbehaviour.

VII.7. RELATIVISTIC DIFFUSION 251

VII.7.1 Left Brownian motion on the Poincare group Pd+1

Related to PSO(1, d) is the Poincare group Pd+1. This group is theanalogue in the present Lorentz-Minkowski set-up of the classical groupof rigid motions. It is the Lie subgroup of GL(d+2) made of the matrices

having the form

(Λ ξ0 1

), with Λ ∈ PSO(1, d), ξ ∈ R1,d (written as a

column), and 0 ∈ R1+d (written as a row). Its Lie algebra is the set of

matrices

(A v

0 0

)∈M(d+ 2), with A ∈ so(1, d) and v ∈ R1,d.

Take k = d , Aj =

(Ej 00 0

)for 1 ≤ j ≤ d , where Ej is the

infinitesimal boost of Section I.3, and A0 =

(0 e0

0 0

).

The solution (Xs) to the linear S.D.E. (VII.1) is thus a left Brownian

motion on the Poincare group Pd+1. Set Xs =

(Λs ξs0 1

). Equation

(VII.1) is then equivalent to :

Λs = 1+

∫ s

0

Λt dWEt + 1

2

d∑j=1

∫ s

0

ΛtE2j dt and ξs =

∫ s

0

Λt e0 dt . (VII.16)

Hence on the one hand (Λs) is the solution of Equation (VII.10), and onthe other hand we have ξs = Λs e0 .

Consider the projection p defined by p

(Λ ξ

0 1

):= (ξ,Λe0). It satis-

fies the criterion of Proposition VII.5.2, since by Equation (VII.16) thisamounts to the irrelevance of % ∈ SO(d) in the law of (% ξt, % ξt = %Λt e0),and the latter is clear from Section VII.6.7, according to which (%Λt e0)is a hyperbolic Brownian motion started at % e0 = e0 .


Corollary VII.7.1.1 The left Brownian motion Xs =

(Λs ξs0 1

)in-

duces (by projection) a diffusion process (ξs, ξs) on R1,d × Hd (in thesense of Definition VII.5.1), such that (ξs) is a hyperbolic Brownian mo-tion. This T 1R1,d-valued diffusion process is a relativistic diffusion. Itsgenerator is the relativistic operator :

L := ξ∂

∂ξ+ 1

2 ∆ξ ≡d∑j=0

ξj∂

∂ξj+ 1

2 ∆ξ .

Note that ξs is parametrized by its arc length. Mechanically, ξsdescribes the trajectory of a relativistic particle of small mass indexed byits proper time, submitted to a white noise acceleration (in proper time).In any fixed Lorentz frame the time coordinate t(s) = (ξs)0 is strictlyincreasing and the velocity

(vjs := (ξs)j/(ξs)0

∣∣ 1 ≤ j ≤ d)

is bounded bythe velocity of light (which equals 1 in this set-up). In particular, theserelativistic trajectories cannot be closed : in the terminology of [H-E],they satisfy the “causality condition”.

VII.7.2 Asymptotic behaviour of the relativistic diffusion

Let us parametrise Hd by the polar coordinates (ρ, φ) ∈ R+×Sd−1 ofProposition III.5.4, and denote by (ρs, φs) the polar coordinates of thehyperbolic Brownian motion (ps) arising in Corollary VII.7.1.1. Recallthat this means : ξs ≡ ps = (ch ρs, φs sh ρs) ∈ Hd.

By Formula (III.11) giving the expression of ∆ in these coordinates,by Proposition VII.6.6.2 defining the spherical Brownian motion and byIto’s Formula, we have :

dρs = dws + d−12 coth ρs ds , dφs =

dΨs

sh ρs, (VII.17)


where we denote by : (Ψs) a spherical Brownian motion on Sd−1, (ws) astandard real Brownian motion, independent of (Ψs).

We know from Proposition VII.6.7.3 that almost surely as s → ∞ ,the radial diffusion (ρs) goes to infinity and the angular coordinate (φs)goes to some random limit φ∞ ∈ Sd−1. Let us be more precise now.

Lemma VII.7.2.1 We have ρs = ws + d−12 s + ηs , for some almost

surely converging (as s→ +∞) process (ηs) . Hence the random variable∫ ∞0

(sh ρu)−2 du is almost surely finite.

Proof By the comparison theorem VII.1.2 (since coth ρs > 1) and byProposition VI.3.3 we have almost surely :

ρs ≥ ρ0 + ws + d−12 s −→ +∞ .

This proves that we almost surely have ρs ≥ (d− 1)s/3 for large enoughs (which is stronger than the mere ρs →∞ already known from Propo-sition VII.6.7.3). Hence, setting ηs := ρs − ws − (d− 1)s/2 , we deduce

that ηs = ηs0+ (d− 1)

∫ s

s0

ds

e2ρs − 1converges almost surely.

Consider a fixed observer located at the spatial origin 0, having thenrelativistic trajectory (τ, 0) ∈ R+×Rd ⊂ R1,d, with proper time τ . Whilethe relativistic diffusion has trajectory ξs =

((ξs)0, (ξs)1, . . . , (ξs)d

)=:(

t(s), ~ξs)∈ R+×Rd. As we already saw in Section II.4, at time τ in the

observer’s frame, the position of the relativistic diffusion is Z(τ) := ~ξs(τ) ,for s(τ) determined by τ = t

(s(τ)

), and the velocity d

dτZ(τ) = φs th ρshas norm < 1 , i.e., less than the velocity of light. Moreover we have thefollowing, which states in particular that the magnitude of the velocityasymptotically becomes that of the light.

Corollary VII.7.2.2 The mean Euclidean velocity Z(τ)/τ goes almostsurely to φ∞ ∈ Sd−1 as τ →∞ .


Proof By Proposition VII.6.7.3 and Lemma VII.7.2.1, and since

limτ∞

s(τ) = +∞ and ddτZ(τ) = φs(τ) th ρs(τ) , we immediately have :

limτ→∞

th ρs(τ) = limτ→∞

√1−

∣∣p0s(τ)

∣∣−2= 1 , and then lim

τ→∞dZ(τ)

dτ= φ∞ ,

almost surely. The result follows at once.

Corollary VII.7.2.2 (after Proposition VII.6.7.3) yields the limit ran-dom direction φ∞ ∈ Sd−1, an asymptotic variable produced by the rel-ativistic diffusion (ξs, ξs). We now obtain another one. Its geometricmeaning is to specify how almost every trajectory of the relativistic dif-fusion goes away to infinity in the random direction φ∞ : actually, byapproaching asymptotically some random affine hyperplane of R1,d, par-allel to the tangent hyperplane to the light cone which contains e0 +φ∞ .

Proposition VII.7.2.3 The random variable 〈ξs , e0 + φ∞〉 convergesalmost surely as s→∞ , yielding a second (strictly positive) asymptoticrandom variable provided by the relativistic diffusion (ξs, ξs), beside φ∞ .

Proof From the above we straightforwardly deduce that :⟨ξs − ξ0, e0 + φ∞

⟩=

∫ s

0

[ch ρt − φt · φ∞ × sh ρt

]dt

=

∫ s

0

e−ρtdt+

∫ s

0

[(φ∞ − φt) · φ∞

]sh ρt dt .

Here, φ1 · φ2 denotes the Euclidean inner product in Rd. It is clear

from Lemma VII.7.2.1 that

∫ s

0

e−ρtdt converges almost surely. Then,

by Formula (VII.17) and Lemma VII.7.2.1 we can use a Sd−1-valuedBrownian motion (Φt), independent of (ws) and then of the radial process

(ρs), to write φ∞ = Φ0 and φt = Φ

[ ∫ ∞t

(sh ρu)−2 du

]. Denoting by ϕ


the angular distance on Sd−1 to Φ0 (taken as north pole), we then have

(φ∞ − φt) · φ∞ = 1− cosϕ

(Φ

[ ∫ ∞t

(sh ρu)−2 du

]).

Now, recall from Proposition III.4.2(iii) that the spherical Laplaciandecomposes according to :

∆Sd−1

=∂2

∂ϕ2+ (d− 2) cotgϕ

∂

∂ϕ+ (sinϕ)−2∆Sd−2

, so that (by Ito’s For-

mula) the so-called Legendre process(ϕt := ϕ Φt ∈ [0, π]

)solves the

stochastic differential equation : dϕt = dwt + d−22 cotgϕt dt, for some

standard real Brownian motion (wt). Hence we almost surely have :

1− cosϕt =

∫ t

0

sinϕs dws + d−12

∫ t

0

cosϕs ds =

∫ t

0

sinϕs dws +O(t)

near 0 . To proceed further, we need the following lemma, relative toan abstract Brownian martingale.

Lemma VII.7.2.4 Set Mt :=

∫ t

0

Hs dBs , for a real Brownian motion

(Bs) and H ∈ Λ such that |Hs| ≤ hs , for some locally bounded deter-ministic (hs).

Proof By Doob’s inequality (Theorem VI.3.2(i)) for any n ∈ N we have :

P

[sup

0≤t≤2−n|Mt| >

(∫ 2−n

0

h2

)1−ε2

]≤(∫ 2−n

0

h2

)ε−12

E[∣∣∣ ∫ 2−n

0

Hs dBs

∣∣∣]

≤(∫ 2−n

0

h2

)ε−12

E1/2

[ ∫ 2−n

0

H2s ds

]≤(∫ 2−n

0

h2

)ε/2= O

(2−n ε/2

),

by the Schwarz inequality, the isometric Ito identity (VI.3), and the hy-pothesis. Hence by the first Borel-Cantelli lemma there exists almost


surely a n0 ∈ N such that for any t ∈ ]0, 2−n0], choosing n ≥ n0 conve-niently, we have :

2−n−1 ≤ t ≤ 2−n ⇒ |Mt| ≤(∫ 2−n

0

h2

)1−ε2

≤(∫ 2t

0

h2

)1−ε2

.

End of the proof of Proposition VII.7.2.3 : Applying Lemma VII.7.2.4

with hs ≡ 1 to the continuous martingale

∫ t

0

sinϕs dws , we deduce that

almost surely12 ϕ

2t ∼ 1− cosϕt = o

(t

1−ε2

)near 0.

This implies that we can apply Lemma VII.7.2.4 again, with hs =o(s

1−ε4

), which (for a convenient ε) yields : almost surely,

1− cosϕt = o

(∫ 2t

0

s1−ε

2 ds

)1−ε2

= o(t

3−4ε+ε2

4

)= o(t

35

)near 0.

Then, by this and by Lemma VII.7.2.1, we almost surely have :∫ ∞0

[(φ∞ − φt) · φ∞

]sh ρt dt = O(1)

∫ ∞0

[ ∫ ∞t

(sh ρu)−2 du

]sh ρt dt

= O(1)

∫ ∞0

[ ∫ ∞t

exp[−2wu − (d− 1)u] du

]3/5

× exp[wt + d−1

2 t]dt

= O(1)

∫ ∞0

[ ∫ ∞t

exp[− (d− 1− ε)u

]du

]3/5

exp[d−1+ε

2 t]dt <∞

(for 0 < ε < 1/11). Hence, the following limit exists indeed almostsurely :

lims→∞

⟨ξs−ξ0, e0 +φ∞

⟩=

∫ ∞0

e−ρtdt+

∫ ∞0

[(φ∞−φt) ·φ∞

]sh ρt dt > 0 .

It is proved in [Bl] that(φ∞ , lim

s→∞

⟨ξs, e0 +φ∞

⟩)actually exhausts the asymptotic σ-field

of the relativistic diffusion.

VII.8. NOTES AND COMMENTS 257

Remark VII.7.2.5 For d ≥ 3 , the Legendre process(ϕt = ϕ Φt ∈ [0, π]

)arising in the

preceding proof(of Proposition VII.7.2.3, as the angular distance of the spherical Brownian

motion (Φt) to a pole P ∈ Sd−1) almost surely never hits 0, π. In other words, for d ≥ 3the rotations % mapping e1 to ±e1 happen to be polar for the SO(d)-valued left Brownianmotion (Xt) of Proposition VII.6.4.1 : we almost surely have Xt e1 6= ±e1 for all t > 0 .

Proof By its very definition, the Legendre process ϕt = d(P,Φt) hits ]0, π[ at arbitrarysmall positive times. Hence it is enough to consider the case of ϕ0 ∈ ]0, π[ . We have to showthat almost surely (ϕt) never hits 0, π, for d ≥ 3 . Recall that (ϕt) satisfies the stochasticdifferential equation dϕt = dwt + d−2

2 cotgϕt dt . Consider then the function defined for

0 < ϕ < π by f(ϕ) :=

∫ ϕ

π/2(sinψ)2−d dψ , and the real stochastic process f(ϕt), almost

surely defined as long as (ϕt) does not hit 0, π. Note that limπf =∞ = − lim

0f . Applying

Ito’s Formula we obtain :

d(f(ϕt)

)= f ′(ϕt) + 1

2 f′′(ϕt) d〈ϕ,ϕ〉t

= (sinϕt)2−d(dwt + d−2

2 cotgϕt dt)

+ 2−d2 (sinϕt)

1−d cosϕt dt = (sinϕt)2−d dwt .

Thence, considering the time change τt := inf

s

∣∣∣∣ ∫ s

0(sinϕu)4−2d du > t

, we see that f

(ϕτt)

is a continuous martingale having quadratic variation t , hence a real Brownian motion Wt(started at f(ϕ0)

), according to Levy’s Theorem VI.5.7 (and Remark VI.5.8). As a conse-

quence, we almost surely have f(ϕs) = W

(∫ s

0(sinϕu)4−2d du

)for all s smaller than the

hitting time T of 0, π by (ϕs). Now, if T were finite then as s T , f(ϕs) would go to +∞if ϕT = π and to −∞ if ϕT = 0 ; whereas by Proposition VI.3.3, W

(∫ s

0(sinϕu)4−2d du

)almost surely cannot go to infinity keeping a given sign. This shows that T is almost surely

infinite, thereby ending the proof.

VII.8 Notes and comments

Linear Stochastic Differential Equations appear as deserving a specifictreatment in [Ha]. Developments followed in the directions of productsof random matrices, see in particular [FgK], [L1], and of stochastic flows,see for example [CE1], [LJW], [CE2], [Ba], [Ku].


Lyapounov exponents are also used in the context of (stochastic ordeterministic) dynamical systems ; see [Os], [Ru], [L1].

The first mathematically convincing treatment of relativistic diffu-sions is due to Dudley [D1], [D2], [D3]. The generalization to Lorentzmanifolds appeared in [FLJ1]. Some other references on this subject are[Bl], [F4], [E], [FLJ2], [BF]. See [Deb] for a physical point of view.

Chapter VIII

Central Limit Theorem forgeodesics

In this chapter we provide a proof of the Sinai Central Limit Theorem,generalized to the case of a cofinite and geometrically finite Kleiniangroup. This theorem, which completes the mixing property establishedin Section V.3, shows that asymptotically geodesics behave chaotically,and yields a quantitative expression of this phenomenon.

The method we use is by establishing such a result first for Brow-nian trajectories, which is easier because of their strong independenceproperties. Then we compare geodesics with Brownian trajectories, bymeans of a change of contour and time reversal. This requires in par-ticular considering diffusion paths on the stable foliation and derivingthe existence of a key potential kernel, using the spectral gap exhibitedin Section V.4 and the commutation relation of Section II.6 (which isrelated to the instability of the geodesic flow, recall Proposition II.3.9).

Fix a cofinite Kleinian group Γ admitting a spectral gap, for examplea geometrically finite one (according to Theorem V.4.2.5).

We shall make a large use of the Lie derivatives on Fd, associated tothe flows.

259

260 CHAPTER VIII. CENTRAL LIMIT THEOREM FOR GEODESICS

Notation We lighten somewhat the notations of Sections I.1.4 andI.4, setting

L0 := LE1and Lj := LEj , for 2 ≤ j ≤ d ,

that is, for any f ∈ C1(Fd) and β ∈ Fd (and for 2 ≤ j ≤ d) :

L0f(β) :=dodsf(β θs) , and Ljf(β) :=

dodsf(β θ+

sej) . (VIII.1)

We need to control also the Lie derivatives associated with so(d). Re-call from Section I.3 that so(d) is spanned by the infinitesimal rotationsEkl = 〈ek, ·〉 el − 〈el, ·〉 ek. Thus we set

Lkl := LEkl , for 1 ≤ k, l ≤ d .

Recall from Formulas (III.4) that (for 2 ≤ j ≤ d) we have :

L0f(βTx,y) = y∂

∂yf(βTx,y), and Ljf(βTx,y) = y

∂

∂xjf(βTx,y). (VIII.2)

Recall from Section V.1 the notation introduced after PropositionV.1.1 : Γ\Fd

(Γ\Hd

)denotes the quotient of Fd

(Hd)

under the leftaction of the Kleinian group Γ, so that we can identify Γ-invariantfunctions on Fd

(Hd)

with functions on Γ\Fd(

Γ\Hd).

Similarly, Fd/SO(d−1) denotes the quotient of Fd under the right actionof the rotation group SO(d − 1), so that we can identify SO(d − 1)-invariant functions on Fd with functions on Fd/SO(d − 1) ≡ T 1Hd ≡Hd × ∂Hd. Recall Section II.2.2.

In the same way, we can identify Γ-left invariant and SO(d − 1)-rightinvariant functions on Fd with functions on the two-sided quotient

Γ\Fd/SO(d − 1), which we shall denote henceforth by M. So that wehave Γ\Hd ≡ Γ\Fd/SO(d) ≡ π0(M).

VIII.1. DUAL AD-VALUED LEFT BROWNIAN MOTIONS 261

VIII.1 Dual Ad-valued left Brownian motions

We return now to the Ad-valued left Brownian motion (Zs) solvingEquation (VII.11) in Section VII.6.5. Recall that it has generator

12 D = 1

2

[ d∑j=2

L2j + L2

0 + (1− d)L0

],

and that according to Section VII.6.7, it projects on Hd under π0 tothe hyperbolic Brownian motion ; precisely, for any β ∈ Fd, π0(β Zs) isa hyperbolic Brownian motion on Hd, started at π0(β) = β(e0).

Recall also that according to Formulas (VII.12) for any t ≥ 0 and forsome R× Rd−1-valued Brownian motion (w,W ) we have :

Zt = Tzt , with yt = ewt−(d−1)t/2 and xt =

∫ t

0

ys dWs . (VIII.3)

We shall also use as a key ingredient (in Section VIII.5) the dualdiffusion process (Z∗s ), which is the Ad-valued left Brownian motionsolving the equation obtained from (VII.11) by changing merely E1

into −E1 . Its generator is

12 D∗ = 1

2

[ d∑j=2

L2j + L2

0 + (d− 1)L0

],

i.e., the adjoint of 12 D with respect to λ , according to Corollary III.3.11.

Denote by P ∗s = exp[(s/2)D∗

]its semi-group, adjoint to the semi-group

(Ps) of (Zs) with respect to λ .

Remark VIII.1.1 (i) For any s , Z∗s and Z−1s have the same law.

(ii) We have D∗F = y1−dD(yd−1 F ) and then P ∗s F = y1−d Ps(yd−1 F ),

for any F ∈ C2b (Ad) and s ≥ 0.


Proof (i) follows at once from Proposition VII.2.11. For (ii), note thatthe coordinate function y makes sense on Ad, and then that applyingFormulas (III.4) we obtain y1−dL0(y

d−1 F ) = L0F + (d− 1)F at once,whence easily D∗F = y1−dD(yd−1 F ) and then the last expression.

As for (Zt) in Formula (VIII.3), there exists a unique zt = (xt, yt) ∈Rd−1×R∗+ such that for some independent Brownian motions w,W andfor any t ≥ 0 :

Z∗t = Tzt , with yt = ewt+(d−1)t/2 and xt =

∫ t

0

ys dWs . (VIII.4)

We shall need at a crucial step (in Section VIII.7, under the form ofRemark VIII.1.3) the following technical lemma, the main ingredient ofwhich is the calculation of the Lie derivatives of the convolution of asmooth function by the laws of (Zs), (Z

∗s ).

Lemma VIII.1.2 For any n ∈ N, both semi-groups (Ps) and (P ∗s ) acton Cn bounded functions on PSO(1, d) which have bounded Lie deriva-tives of order ≤ n . Moreover this result remains valid when Lj-deriva-tives only are considered : (Ps) and (P ∗s ) act on functions on Ad havingbounded L0,L2, . . . ,Ld-derivatives.

Proof 1) By Proposition I.3.2 and Formula (I.15), we have to considerthe Lie derivatives L0 , Lj and L1j for 2 ≤ j ≤ d , and Ljk for 2 ≤j < k ≤ d . Fix some f having bounded derivatives. Up to replacethe function f by its left-translated fγ := f(γ ·), for any given γ ∈PSO(1, d), it is enough to consider the Lie derivatives at the unit 1 ofPSO(1, d). Using Formula (VIII.3), for any bounded function f on Ad

and any t ≥ 0 we have :

Ptf(1) = E[f(Zt)] = E[f(T(xt,log yt))

]= E

[f(θ+

xtθlog yt)

]. (VIII.5)

By (VIII.4) we similarly have :

P ∗t f(1) = E[f(Z∗t )] = E[f(T(xt,log yt))

]= E

[f(θ+

xtθlog yt)

]. (VIII.6)

DUAL Ad-VALUED LEFT BROWNIAN MOTIONS 263

We deduce from Formulas (VIII.3), (VIII.1), (VIII.5) and (I.16) :

L0Ptf(1) =dods

E[f(θs θ

+xtθlog yt)

]= E

[dodsf(Zt θ

+(es−1)xt/yt

θs)

]

= E[L0f(Zt) +

d∑j=2

xjtytLjf(Zt)

]= PtL0f(1) + E

[y−1t

d∑j=2

xjt Ljf(Zt)].

Similarly, for 2 ≤ j ≤ d we have :

LjPtf(1) =dods

E[f(θ+

sejθ+xtθlog yt)

]= E

[dodsf(Zt θ

+sej/yt

)

]= E

[y−1t Ljf(Zt)

],

and

L1jPtf(1) =dods

E[f(esE1j Zt)

]= E

[dodsf(Zt Ad(Z−1

t )(esE1j))]

= E[dodsf(Zt exp

[sAd(Z−1

t )(E1j)])]

.

2) By Lemma III.3.9, for any t ≥ 0 and 2 ≤ j ≤ d , on the one handwe have :

Ad(Z−1t )(E1j)

= ytE1j + xjt E1 +d∑

k=2

xkt Ekj −|xt|2 + y2

t − 1

2 ytEj +

xjtyt

d∑k=2

xkt Ek ,

whenceL1jPtf(1)

= E

[(ytL1j+x

jt L0−

|xt|2 + y2t − 1

2 ytLj+

xjtyt

d∑k=2

xktLk+d∑

k=2

xktLkj)f(Zt)

];


and on the other hand, for 2 ≤ i < j ≤ d :

Ad(Z−1t )(Eij) = Eij −

xitytEj +

xjtytEi ,

whence

LijPtf(1) = Pt[Lijf ](1) + E

[xjtytLif(Zt)−

xitytLjf(Zt)

].

3) Of course, we have the same formulas for L0P∗t f , LjP ∗t f , L1jP

∗t f ,

LkjP ∗t f , by Formula (VIII.6), merely changing (xt, yt) into (xt, yt).Hence the first sentence of the lemma will follow from the integrabilityof the random variables : xjt/yt , y

−1t , xjt , x

jt x

kt /yt , and similarly with

(xt, yt) instead of (xt, yt), and then from the integrability of the randomvariables :

y−2t , y2

t , (xjt xkt )

2 , y−2t , y2

t , (xjt xkt )

2 .

Now Formulas (VIII.3) and (VIII.4) imply at once the integrability ofynt , y

nt , for all n ∈ Z, and by Proposition II.1.1 we have :

E[|xjtxkt |2

]≤ 1

2

(E[|xjt |4

]+ E

[|xkt |4

])≤ C2 E

[∣∣∣ ∫ t

0

y2s ds∣∣∣2]

≤ C2 t E[ ∫ t

0

y4s ds

]<∞ ,

and similarly for E[|xjt xkt |2

]. This establishes the first sentence of the

lemma.

4) To prove the second sentence of the lemma, we deal with the secondderivatives in the same way. Thus from the above, for 2 ≤ j, k ≤ d , wesuccessively obtain :

L20Ptf(1) =

DUAL Ad-VALUED LEFT BROWNIAN MOTIONS 265

PtL20f(1)+E

[y−1t

d∑j=2

xjt(LjL0+L0Lj)f(Zt)

]+E[y−2t

d∑j,k=2

xjtxkt LjLkf(Zt)

],

L0LjPtf(1) = E[y−1t L0Ljf(Zt)

]+ E

[y−2t

d∑k=2

xjtxkt LkLjf(Zt)

],

LjL0Ptf(1) = E[y−1t LjL0f(β Zt)

]+ E

[y−2t

d∑k=2

xkt LjLkf(Zt)

],

LjLkPtf(1) = E[y−2t LjLkf(Zt)

],

and so on with the rotational derivatives. Thus we see that the claimabout second order derivatives will follow from the integrability of therandom variables : (xjt x

kt )

4, (xjt xkt )

4. Very similarly, the lemma willfollow from the integrability (for any n ∈ N) of the random variables :(xjt)

2n , (xjt)2n. Finally, by Proposition VI.5.9 this again amounts to the

already quoted integrability of ynt , ynt .

We still denote the semi-groups corresponding to the right action of(Zs), (Z

∗s ) on Γ\Fd by (Ps), (P

∗s ). Using the notation : fβ(γ) := f(βγ)(

for any β ∈ Fd, γ ∈ PSO(1, d)), we thus merely have :

Ptf(β) = Ptfβ(1) ,

for any non-negative measurable function f on Γ\Fd and any β ∈ Fd,t ≥ 0 .

Remark VIII.1.3 By the previous lemma VIII.1.2, both semi-groups(Ps) and (P ∗s ) act on Cn bounded functions on Γ\Fd which have boundedLie derivatives of order ≤ n . Moreover this result remains valid whenLj-derivatives only are considered : (Ps) and (P ∗s ) act on functions onΓ\Fd having bounded L1, . . . ,Ld-derivatives.


Lemma VIII.1.4 (i) The hyperbolic heat semi-group (Qs) acts on Γ-left invariant functions on Hd : we have Qs[F π0](γ p) = Qs[F π0](p)for any p ∈ Hd, γ ∈ Γ and test-function F on Γ\Fd.(ii) The relative volume measure dΓp (recall Proposition V.1.1(ii)) isinvariant under the hyperbolic Brownian semi-group.

Proof (i) For any β ∈ Fd such that β0 = p , we have :

PsF (γβ) = E[F (γβZs)

]= E

[F (βZs)

]= PsF (β), which by projection

under π0 directly yields Qs[F π0](γ p) = Qs[F π0](p) .

(ii) Specialising Proposition V.1.1(iv) to g = Zs , where (Zs) is thePSO(1, d)-valued left Brownian motion solving Equation (VII.11), wehave :∫f(β Zs) λ

Γ(dβ) =

∫f dλΓ. Taking expectation, this yields :∫

Psf dλΓ =

∫f dλΓ, where (Ps) denotes the Markovian semi-group

associated to (Zs). Let (Qs) denote the hyperbolic heat semi-group ofSection VII.6.7. Using Formula (VII.14), for any Γ-left invariant test-function h on Hd we obtain :∫

Qsh(p) dΓp =

∫(Qsh) π0 dλ

Γ =

∫Ps(h π0) dλ

Γ

=

∫h π0 dλ

Γ =

∫h(p) dΓp .

We shall establish that (Qs) is actually self-adjoint with respect to dΓp,see Theorem VIII.2.4 below.

VIII.2. TWO DUAL DIFFUSIONS 267

VIII.2 Two dual diffusions

We now use the right action of PSO(1, d) on Fd and Γ\Fd, andProposition V.1.1.

Definition VIII.2.1 Set µ := λΓ/

covol(Γ), and fix a Γ\Fd-valued ran-dom variable ξ0 , independent from (Zs), (Z

∗s ) and having law µ . Set

ξs := ξ0 Zs and ξ∗s := ξ0 Z∗s , for any s ≥ 0 .

Proposition VIII.2.2 Both diffusions (ξs) and (ξ∗s) are stationary

(with law µ) on Γ\Fd, and dual of each other : for any test-functionsϕ , ψ on Γ\Fd and any s ≥ 0 , we have

E[ϕ(ξs)ψ(ξ0)

]= E

[ϕ(ξ∗0)ψ(ξ∗s)

].

Proof The stationarity is clear from the PSO(1, d)-right-invariance ofthe Liouville measure

(recall Proposition V.1.1(iv)

). Then using Remark

VIII.1.1, Proposition V.1.1(iv) and Definition VIII.2.1, we have :

E[ϕ(ξ∗0)ψ(ξ∗s)

]= E

[ϕ(ξ0)ψ(ξ∗s)

]= E

[ϕ(ξ0)ψ(ξ0 Z

−1s )]

= E[ ∫

Γ\Fdϕ(ξ)ψ(ξ Z−1

s )µ(dξ)

]= E

[ ∫Γ\Fd

ϕ(ξ Zs)ψ(ξ)µ(dξ)

]= E

[ϕ(ξs)ψ(ξ0)

]= E

[ϕ(ξs)ψ(ξ∗0)

].

Remark VIII.2.3 By Section VII.6.7 and Remark VII.5.3 we havePt(f π0) = (Qtf) π0 , for any bounded f on Hd and any t ≥ 0.In the same way, by Lemmas VIII.1.3 and VIII.1.4(i) for any boundedmeasurable h on Γ\Hd we have Pt(h π0) = (Qth) π0 , and then


ddt Qth = 1

2 ∆Qth = 12 Qt ∆h , i.e., Formulas (VII.15) remain valid on the

quotient Γ\Hd.

Moreover, again by Section VII.6.7 and Formula (VII.14) especially, forany z ∈ Γ\Hd and β ∈ Γ\Fd such that β0 = z , zt := (β Zt)0 definesa diffusion on Γ\Hd, such that Qtf(z) = E

[f(zt)

]for any test-function

f on Γ\Hd. Such (zt) is a hyperbolic Brownian motion (started at z)on Γ\Hd ≡ π0(M).

Thus the diffusion (ξs) projects under π0 to the stationary hyperbolicBrownian motion on Γ\Hd.

We deduce the following important property from is analogue inL2(dp), which was established in Theorem VII.6.7.2.

Theorem VIII.2.4 The hyperbolic Brownian semi-group (Qs) is self-adjoint in L2(dΓp).

Proof We have to quotient (on the left hand side) by Γ. Fixing afundamental domain D and using Remark VIII.2.3, we have :∫

D

Qsf × h dp =

∫Hdf Qs(h1D) dp =

∫Fdf π0 ×Qs(h1D) π0 dλ

=

∫Fdf π0 Ps

(h π0 1π−1

0 (D)

)dλ

= E[ ∫

Fdf π0(β)h π0(βZs) 1π−1

0 (D)Z−1s

(β) dλ(β)

].

As π−10 (D)Z−1

s is almost surely another fundamental domain for theaction of Γ on Fd, using the Γ-invariance of f, h and λ , we get :∫

Qsf × h dΓp =

∫D

Qsf × h dp

VIII.3. SPECTRAL GAP ALONG THE FOLIATION 269

= E[ ∫

Fdf π0(β)h π0(βZs) 1π−1

0 (D)(β) dλ(β)

]=

∫π−1

0 (D)

f π0 Ps(h π0) dλ =

∫D

f Qsh dp =

∫f Qsh d

Γp .

By Theorem V.4.2.5, the hyperbolic Brownian semigroup (Qs) satis-fies a Poincare inequality : for any h ∈ L2(dΓp) having zero mean andany s > 0, we have :

d

ds

∫|Qsh|2(p) dΓp =

∫D

Qsh∆Qsh = −∫D

γ(Qsh)

≤ −C−1D

∫|Qsh|2(p) dΓp ,

since by Proposition V.1.1

∫Qsh(p) dΓp = 0 . This entails the following.

Corollary VIII.2.5 The hyperbolic Brownian semigroup (Qs) admitsa spectral gap (hence is mixing) in L2(dΓp) : for any h ∈ L2(dΓp) havingzero mean and any s ≥ 0 we have :∫

|Qsh|2(p) dΓp ≤ e−C−1D s

∫h2(p) dΓp .

VIII.3 Spectral gap along the foliation

In Section V.4 we established the existence of a Poincare inequality,and then of a spectral gap, at the level of the hyperbolic space, actuallyin a cofinite fundamental domain. We want now to extend this cru-cial spectral gap property to the level of the tangent space, and moreprecisely, to the foliation defined by the geodesic and horocyclic flows.


d

d

Figure VIII.1: extremity u(ρ) of the geodesic determined from e0 by ρ ∈ SO(d)

This will be done using appropriate norms. In contrast with the Brow-nian case, the exponential decay does not hold for all L2 functions, butfor the rotationally Holderian ones, due to the fact that the diffusionevolves along a foliation. This will immediately imply the existence of apotential kernel for the foliated Brownian semi-group (Pt).

We shall use here the Poincare coordinate u(%) considered in TheoremII.6.1, which is also the extremity of the half-geodesic (e0, %

′e0e1) θR+

⊂ Hd

determined by the tangent vector dods %(e0 + s e1) at its starting point

% e0 = e0 . See Figure VIII.1 and recall Proposition II.6.2 : decomposing% = Rα,σ % with α ∈ [0, π], σ ∈ Sd−2 and % ∈ SO(d − 1), we haveu(%) = cotg (α2 )σ , and the above vector is % e1 = (cosα)e1 + (sinα)σ.

We need to consider the following projection of generic functions onHd × ∂Hd onto functions on Hd.

Definition VIII.3.1 For any non-negative Borelian function F on


Hd × ∂Hd, and any p ∈ Hd, set

F (p) :=

∫∂Hd

F (p, η)µp(dη) ,

where µp is the harmonic measure at p (recall Definition III.6.1.1). Forany non-negative Borelian function H on Fd, any (p, η) ∈ Hd×∂Hd andany β ∈ π−1

1 (p, η), set :

H(p, η) :=

∫π−1

1 (p,η)

H d%(p,η) =

∫SO(d−1)

H(β %) d% , and H := H ,

where d%(p,η) ≡ d% denotes the normalised Haar measure on π−11 (p, η) ≡

SO(d− 1).

Note that by Propositions III.6.2.4 and III.6.2.2(ii), (for the same func-tions F,H as above) we have :∫

HdF (p) dp =

∫Hd×∂Hd

F d(λ π−11 ) and

∫HdH(p) dp =

∫FdH dλ .

Note also that, thanks to the geometric property of harmonic mea-sures (recall Remark III.6.1.3), if F is Γ-invariant, then F is Γ-inva-riant too. Similarly, if H is Γ-invariant, then H is too. And then, forΓ-invariant functions F,H recalling Proposition V.1.1 we have :∫

F (p) dΓp =

∫F d(λΓ π−1

1 ) and

∫H(p) dΓp =

∫H dλΓ .

Definition VIII.3.2 Given r > 0 , a Borelian function F on Hd×∂Hd

such that

|||F |||r := supβ∈Fd, %∈SO(d) \ SO(d−1)

∣∣F π1(β%)− F π1(β)∣∣× `(%)−r <∞

is said to be rotationally r-Holderian. Here `(%) denotes the distancefrom % to SO(d − 1)

(which can be replaced by

∣∣ sin(α2 )∣∣ or 1

|u(%)|+1,

recall Proposition II.6.2).


Remark VIII.3.3 The semi-group (Pt) of (ξt) acts not only on

L∞(Γ\Fd, λΓ), but makes sense on L∞(M, λΓ

)as well : (Pt) does act on

SO(d− 1)-invariant functions, that is, on functions on Γ\(Hd × ∂Hd).

Moreover, it is contracting in L2(Γ\Fd, λΓ) : ‖PtF‖2 ≤ ‖F‖2 if

t ≥ 0 and F ∈ L2(Γ\Fd, λΓ).

Proof Indeed, since ξs = ξ0Zs and (Zs) is Ad-valued, this is clear fromthe commutativity between SO(d−1) and Ad : for any positive Borelianfunction H on Fd and any β ∈ Fd, % ∈ SO(d− 1), we have

PtH π1(β %) = E[H π1(β %Zt)

]= E

[H π1(β Zt %)

]= E

[H π1(β Zt)

]= PtH π1(β).

And by Schwarz inequality and right-invariance of λ :

‖PtF‖2L2(Γ\Fd,λΓ)

=

∫Γ\Fd|PtF |2dλΓ ≤

∫Γ\Fd

Pt(F2)dλΓ = ‖F‖2

L2(Γ\Fd,λΓ).

Theorem VIII.3.4 For any F ∈ L∞(M, λΓ

)(hence bounded Γ-inva-

riant and SO(d− 1)-invariant), rotationally r-Holderian and such that∫F dλΓ = 0 , there exist C = C(d) and δ = δ(r,Γ) > 0 such that

‖PtF‖L2(λΓ) ≤ C(‖F‖∞ + |||F |||r

)e−δ t for all t ≥ 0 .

Proof 1) Using Formula (VIII.3), Definition III.6.1.1, and TheoremII.6.1, we have for any p ∈ Hd, β ∈ π−1

0 (p), and t ≥ 0 :

PtF (p) =

∫∂Hd

PtF (p, η)µp(dη) =

∫SO(d)

E[F π1(β % Tzt)

]d%


= cd

∫[0,π]×Sd−2

E[F π1(βRα,σ Tzt)

](sinα)d−2dα dσ(

by Proposition III.3.2, with cd := Γ(d/2)√π Γ((d−1)/2)

)= cd

∫[0,π]×Sd−2

E[F π1(β Tz′tRα′t,σ

′t)]

(sinα)d−2 dα dσ

by the commutation relation of Theorem II.6.1,

with z′t = (x′t, y′t) ∈ Rd−1 × R∗+ such that Tz′te0 = Rα,σ Tzte0 , and with

u(Rα′t,σ′t) =

(u(Rα,σ)− x′t

)/y′t .

Then, Section VII.6.7 ensures that Tz′te0 is a hyperbolic Brownian mo-tion as well as Tzte0 . Hence we can perform in the above integral thechange of Brownian motion Tz′te0 7→ Tzte0 , which is equivalent to chang-ing (z′t) into (zt). We get thus :

PtF (p) = cd

∫[0,π]×Sd−2

E[F π1(β TztRαt,σt)

](sinα)d−2dα dσ ,

with u(Rαt,σt) =(u(Rα,σ) − xt

)/yt . By Proposition II.6.2, the latter

reads : cotg (αt2 )σt =(cotg (α2 )σ − xt

)/yt .

The above formula implies that for any (p, η) ∈ Hd × ∂Hd and anyβ ∈ π−1

1 (p, η) we have :

(PtF − PtF )(p, η)

= cd

∫[0,π]×Sd−2

E[F π1(β Zt)− F π1(β ZtRαt,σt)

](sinα)d−2 dα dσ .

2) Fix 0 < ε < 1/2 , and for t ≥ 0 and % ≡ Rα,σ ∈ SO(d), set :

E1t := 1

yt>e−(d−1−ε) t/2 ; E2

t := (1− E1t )× 1|u(%)−xt| ≤ yt eε t/2

;

E3t := (1− E1

t − E2t ) , and for 1 ≤ j ≤ 3 :

Ajt ≡ Aj

t(β)


:= cd

∫[0,π]×Sd−2

E[(F π1(β Zt)−F π1(β ZtRαt,σt)

)×Ej

t

](sinα)d−2dα dσ,

so that (PtF − PtF )(p, η) = A1t + A2

t + A3t . Recall that (zt) satisfies

Equations (VIII.3).

Then on the one hand we have :∥∥Ajt

∥∥2

L2(λΓ)≤ 4 ‖F‖2

∞ ×∫

SO(d)

E[Ejt

]d% ,

so that∥∥A1t

∥∥2

L2(λΓ)≤ 4 ‖F‖2

∞ × P[wt > ε t/2] = ‖F‖2∞ ×O

(e−ε

2 t/8).

On the other hand we have :∥∥A2t

∥∥2

L2(λΓ)≤ 4 ‖F‖2

∞

∫SO(d)

P[|u(%)− xt| ≤ e−(d−1−2ε) t/2

]d%

= 4 ‖F‖2∞ cd E

[∫[0,π]×Sd−2

1|cotg (α2 )σ−xt| ≤ e−(d−1−2ε) t/2

(sinα)d−2 dα dσ

]= 4 ‖F‖2

∞ cd E[∫

Rd−11|u−xt| ≤ e−(d−1−2ε) t/2

( 21+|u|2

)d−2

du

]≤ 2d ‖F‖2

∞ cd

∫Rd−1

1|u| ≤ e−(d−1−2ε) t/2

du = ‖F‖2∞×O

(e−(d−1)(d−1−2ε) t/2

).

Finally we use Theorem II.6.1, Proposition II.6.2 and the Holderian hy-pothesis made on F , to handle the third term as follows :

|A3t | ≤ cd

∫[0,π]×Sd−2

E[∣∣F π1(βZt)−F π1(βZtRαt,σt)

∣∣ 1|u(Rαt,σt )|>eεt/2

] sind−2α dα dσ

≤ supβ∈Fd, %∈SO(d),|u(%)|>eε t/2

∣∣∣F π1(β)− F π1(β%)∣∣∣

= supβ∈Fd, %∈SO(d),`(%) =O(e−ε t/2)

∣∣∣F π1(β)−F π1(β%)∣∣∣ = |||F |||r×O

(e−ε r t/2

).

VIII.4. RESOLVENT KERNEL AND CONJUGATE FUNCTIONS 275

3) So far, we have shown the existence of C = C(d), δ = δ(r) > 0such that for all t ≥ 0 :

‖PtF − PtF ‖L2(λΓ) ≤ C(‖F‖∞ + |||F |||r) e−δ t.

Finally, on the one hand, by the above and by Remarks VIII.2.3 andVIII.3.3 we have :∥∥PtF∥∥L2(λΓ)

≤∥∥Pt/2(Pt/2F − Pt/2F )∥∥L2(λΓ)

+∥∥Pt/2(Pt/2F )∥∥L2(λΓ)

≤∥∥Pt/2F − Pt/2F ∥∥L2(λΓ)

+∥∥Qt/2

(Pt/2F

)∥∥L2(dΓp)

≤ C(‖F‖∞ + |||F |||r) e−δ t/2 +∥∥Qt/2

(Pt/2F

)∥∥L2(dΓp)

,

and on the other hand by Propositions III.6.2.4 and V.1.1 we have :∫Pt/2F (p)dΓp =

∫Pt/2F (p, η)µp(dη)dΓp =

∫Pt/2Fdλ

Γ =

∫FdλΓ = 0.

We conclude by using the spectral gap property of the Brownian semi-group (Qt) in L2(dΓp) (recall Corollary VIII.2.5) : there exists δ′ =δ′(Γ) > 0 such that for any t ≥ 0 :

‖Qt/2(Pt/2F )‖L2(dΓp) ≤ e−δ′t ‖Pt/2F ‖L2(dΓp)

= e−δ′t ‖Pt/2F‖L2(λΓ) ≤ e−δ

′t ‖F‖L2(λΓ) .

VIII.4 Resolvent kernel and conjugate functions

In this section we introduce a resolvent (potential) kernel, which al-lows to exhibit conjugate functions to a given function f on Hd× ∂Hd,provided it has some regularity. This construction will be crucial be-low, to compare geodesics to Brownian paths by means of a contourdeformation.

We begin with a simplified theory of differential 1-forms on Fd, con-taining just what is necessary to implement our contour deformation.


VIII.4.1 Differential 1-forms on Fd

Definition VIII.4.1.1 Let us call 1-form any C1 map from Fd intothe dual Lie algebra so(1, d)∗, and longitudinal 1-form any C1 map fromFd into the dual Lie subalgebra τd

∗ (recall from Section I.4 that τd isthe Lie algebra of Ad).

Using the canonical basis (E1, E2, . . . , Ed) of τd , any longitudinal 1-form ω identifies with a system (ω0, ω2, . . . , ωd) of C1 functions on Fd,

writing : ω = ω0E∗1 +d∑j=2

ωj E∗j .

Definition VIII.4.1.2 A longitudinal 1-form ω is closed if and onlyif (writing it as above) (Li − δ0i)ω

j = Lj ωi for 0 ≤ i < j ≤ d .

Proposition VIII.4.1.3 If a longitudinal 1-form ω is closed, then forany β ∈ Fd, Tx,y ∈ Ad and any C1 path γ ≡ (Txs,ys)0≤s≤t from 1 to Tx,y ,the line integral∫

βγ[0,t]

ω :=

∫ t

0

ω0(βTxs,ys) y−1s dys +

d∑j=2

∫ t

0

ωj(βTxs,ys) y−1s dxjs (VIII.7)

does not depend on the choice of the path γ . Moreover, setting zs :=

(xs, ys), we have

∫βγ[0,t]

ω =

∫ t

0

ω(βTzs)(T−1zsTzs+ds

).

Proof By Proposition I.4.3 we have

T−1xs,ys

Txs+ds,ys+ds = Txs+ds−xsys

,ys+dsys

= y−1s

( d∑j=2

xjs Ej + ysE1

)ds+ o(ds),

whence ∫βγ[0,t]

ω =

∫ t

0

ω(βTxs,ys)(T−1xs,ys

Txs+ds,ys+ds

).

VIII.4. RESOLVENT KERNEL AND CONJUGATE FUNCTIONS 277

Now by (VIII.2) for 0 ≤ i < j ≤ d we have :

∂

∂y

[ωj(βTx,y)y

−1] = y−2[L0ω

j − ωj](βTx,y)

and∂

∂xj[ωi(βTx,y)y

−1] = y−2Ljωi(βTx,y),

so that the closedness of ω implies that of the 1-form ωβ defined on Rd

by ωβ(x, y) := ω0(βTx,y) y−1dy +

d∑j=2

ωj(βTx,y) y−1dxj.

Hence applying Stokes Theorem to ωβ yields the result.

VIII.4.2 Lift of f to the 1-form ωf

Definition VIII.4.2.1 1) Let us introduce the resolvent kernel, forany q ∈ N∗ and any bounded measurable function φ on Fd :

U qφ(β) :=

∫ ∞0

e−qt φ(β θt) dt . (VIII.8)

2) For any Γ-invariant and SO(d − 1)-invariant bounded Borelian

function f on Fd such that

∫f dλΓ = 0 and the derivatives Ljf,LjLkf

(for 2 ≤ j, k ≤ d) exist and are bounded, let us set (for 2 ≤ j ≤ d) :

fj := −U1Ljf , and ωf := f E∗1 +d∑j=2

fj E∗j . (VIII.9)

Lemma VIII.4.2.2 For f as in Definition VIII.4.2.1 above, the 1-form ωf is closed, with bounded coefficients f, fj . Furthermore, for2 ≤ j, k ≤ d , the derivatives Ljfk = U2LjLkf and L0fk = Lkf + fkexist and are bounded on Fd.


Proof The commutation relation (I.16) implies for 2 ≤ j ≤ d :

Lj U qφ(β) =dods

∫ ∞0

e−qt φ(β θ+sejθt) dt =

dods

∫ ∞0

e−qt φ(β θt θ

+se−tej

)dt

= U q+1Ljφ (β) ,

provided that φ has bounded Lj-derivatives. On the other hand forbounded φ we have :

L0 U qφ(β) =

∫ ∞0

e−qtdodtφ(β θt) dt = q U qφ(β)− φ(β) .

Applying this to φ = −Lkf , we obtain the existence and boundednessof each Ljfk . Moreover, for 2 ≤ j, k ≤ d, since the matrices θ+

sejand

θ+sek

commute, on the one hand we have Ljfk−Lkfj = U2[Lk,Lj]f = 0 ;and on the other hand L0fk − Lkf − fk = 0 . This yields the result,according to Definition VIII.4.1.2.

VIII.5 Contour deformation

In this section we take advantage of the closed form ωf exhibited in

the preceding section, to change the integration path in1√t

∫ t

0

f(β θs) ds :

we substitute the diffusion path ξ[0, t′] :=ξs∣∣ 0 ≤ s ≤ t′ := 2t/(d− 1)

for the geodesic path β[0, t] := β θs | 0 ≤ s ≤ t.In this contour deformation two residual terms appear, that we prove tobe asymptotically negligible. Recall that the diffusion (ξs) was intro-duced in Definition VIII.2.1, and has stationary law µ = covol(Γ)−1λΓ,proportional to λΓ (recall Proposition VIII.2.2).

Precisely, the aim of this section is to establish the following.

VIII.5. CONTOUR DEFORMATION 279

Theorem VIII.5.1 Let f be any Γ-invariant and SO(d−1)-invariant

bounded Borelian function on Fd, such that

∫f dλΓ = 0 and its deriva-

tives Ljf,L2jf (for 2 ≤ j ≤ d) are bounded. Consider the associated

1-form ωf as in Definition VIII.4.2.1. Then for any real a we have :

limt→∞

∫exp

(a√−1√t

∫ (d−1)t2

0

f(β θs) ds

)µ(dβ)

−E[exp

(− a√−1√t

∫ξ[0,t]

ωf)]

= 0 .

Proof 1) Using the geodesic invariance of the measure λΓ (recallProposition V.1.1(iv)), and then of the law µ (of Definition VIII.2.1),for any t > 0 we have :∫

exp

(a√−1√t

∫ (d−1)t2

0

f(β θs) ds

)dµ(β) =

∫exp

(a√−1√t

∫ β θ (d−1)t2

β

ωf)µ(dβ) ,

We write now the change of contour, using Proposition VIII.4.1.3 andLemma VIII.4.2.2 ; we have :∫ β θ (d−1)t

2

β

ωf =

∫ β Txt,yt

β

ωf −∫ β Txt,yt

β T0,yt

ωf −∫ β T0,yt

β θ (d−1)t2

ωf .

Here zt = (xt, yt) ∈ Rd−1 × R∗+ is given by Formula (VIII.4), in whichthe expression of yt explains why we use (Z∗t ) instead of (Zt)

(recall

Formulas (VIII.5) and (VIII.6)). See Figure VIII.2.

2) Since each fj is bounded, by Formula (VIII.7) for any t > 0 (andfor some positive constant C) we have :∫ β Txt,yt

β T0,yt

ωf = y−1t

d∑j=2

∫ 1

0

xjt × fj(β θ+

s xjt ejθlog yt

)ds ≤ C

d∑j=2

|xjt |/yt .


Then by Formula (VIII.4) and Corollary VI.2.3(4) we have

|xjt |/yt =

∫ t

0

ews−wt+(d−12 )(s−t) dW j

s =

∫ t

0

ewt−s−wt−(d−12 )s dW j

t−s

law≡∫ t

0

e−ws−(d−12 )s dW j

s

law≡ xjt −→ xj∞ ∈ R ,

the latter holding almost surely by the proof of Proposition VII.6.7.3.

Hence, uniformly with respect to β ∈ Fd, t−1/2

∫ β Txt,yt

β T0,yt

ωf goes to 0 in

probability as t→∞ . However xj∞ /∈ L2 (for d = 2 or 3).

3) Then for any t > 0 we have :∫ β T0,yt

β θ (d−1)t2

ωf =

∫ wt+(d−1)t/2

(d−1)t/2

f(β θs) ds =

∫ wt

0

f(β θs+(d−1)t/2

)ds ,

and then :

E

[∫exp

[a√−1√t

∫ β T0,yt

β θ (d−1)t2

ωf]µ(dβ)

]

= E

[∫exp

[a√−1√t

∫ wt

0

f(β θs+(d−1)t/2

)ds

]µ(dβ)

]

= E

[∫exp

[a√−1√t

∫ w1

√t

0

f(β θs) ds

]µ(dβ)

]t→∞−−−→ E

[∫exp

[a√−1 w1

∫fdµ

]µ(dβ)

]= 1 ,

by ergodicity (recall Theorems V.3.1 and V.2.3), and since

∫f dµ = 0.

This means that under the law µ ⊗ P, t−1/2

∫ β T0,yt

β θ (d−1)t2

ωf goes to 0 in

probability as t→∞ .

DIVERGENCE OF ωf 281

Figure VIII.2: change of contour : proof of Theorem VIII.5.1

4) So far, and since Txt,yt = Z∗t , we have proved that∫exp

(a√−1√t

∫ (d−1)t/2

0

f(β θs) ds

)µ(dβ)−E

[∫exp

(a√−1√t

∫ β Z∗t

β

ωf)µ(dβ)

]

goes to 0 as t → ∞ . Finally, performing the time reversal allowed byProposition VIII.2.2 we have :

E[∫

exp

(a√−1√t

∫ β Z∗t

β

ωf)µ(dβ)

]= E

[exp

(a√−1√t

∫ ξ∗t

ξ∗0

ωf)]

= E[exp

(a√−1√t

∫ ξ0

ξt

ωf)]

= E[exp

(− a√−1√t

∫ξ[0,t]

ωf)]

.


VIII.6 Divergence of ωf

Fix a SO(d− 1)-invariant bounded function f on Fd, with bounded

Holderian derivatives of order 1 and 2, and such that

∫f dλΓ = 0 .

Lemma VIII.6.1 The potentials U1f and U2f are bounded and

Holderian on Fd.

Proof The boundedness is obvious from (VIII.8). Fix an Holder ex-ponent for f , say 0 < r < 1 . Let us consider a rotation ϕs :=

exp

[s

∑1≤k<j≤d

akjEkj

], for s > 0 and fixed real akj . For q ≥ 1 and

β ∈ Fd, we have :

s−r∣∣∣U qf(βϕs)− U qf(β)

∣∣∣ ≤ ∫ ∞0

e(r−q)t(set)−r∣∣∣ f(β ϕs θt)− f(β θt)

∣∣∣ dt ,which is bounded with respect to (s, β) if (set)−r

∣∣∣ f(βϕsθt) − f(β θt)∣∣∣

or equivalently s−r∣∣∣f(β θ−t ϕse−t θt)− f(β)

∣∣∣ is bounded with respect to

(s, β) and t ≥ 0 .

Now this will be granted if we prove that `(θ−t ϕse−t θt

)= O(s), uni-

formly with respect to t , ` denoting the distance to SO(d− 1)(whose

restriction to SO(d) was already introduced in Definition VIII.3.2, whereit could be replaced by 1

|u(%)|+1, according to Proposition II.6.2

).

Then since Ekj commutes with θt for k, j ≥ 2 , it is sufficient to considerthe case of ϕs = exp[sE1j] = Rs,ej (for 2 ≤ j ≤ d , recalling PropositionIII.3.2).

Now by Theorem II.6.1, we have : `(θ−t ϕs θt) = `(θ−t Tx,yRα,σ), withTx,ye0 = ϕs θt e0 and by Proposition II.6.2 :

u(R−α,σ) = u(R−1α,σ) = u(ϕ−1

s ) e−t = u(R−s,ej) e−t = −e−tcotg (s/2) ej .


By Formula (I.18),

Tx,y e0 = ϕs θt e0 = (ch t)e0 + (sh t cos s)e1 + (sh t sin s)ej

is equivalent to : x = y sh t sin s ej ,

2y sh t cos s = y2 +(y sh t sin)2−1 , and 2y ch t = y2 +(y sh t sin)2 +1 ,

whence y = (ch t − sh t cos s)−1 and x = sh t sin sch t−sh t cos s ej . Hence we

obtain :

`(θ−t ϕse−t θt) = `(θ−t Tx,yRα′,σ′) = `(Te−tx,e−tyRα′,σ′)

≤ dist((e−tx, e−ty); (0, 1)

)+ `(Rα′,σ′)

= O(∣∣∣ e−t(sh t) sin(se−t)

ch t−sh t cos(se−t)

∣∣∣+∣∣∣ e−t

ch t−sh t cos(se−t) − 1∣∣∣+[1 + e−t

∣∣cotg (se−t

2 )∣∣]−1

).

Since for 0 ≤ s ≤ 2π we have(ch t− sh t cos(se−t)

)−1= et

(1 +O(s2)

),

this shows that U qφ is rotationally Holderian on Fd.The Holder property in the geodesic direction is clear since it does notneed any commutation. Finally we obtain the Holder property in thehorocyclic directions, using similarly

s−r∣∣∣U qf(βθ+

s ej)− U qf(β)

∣∣∣≤∫ ∞

0

e(r−q)t supβ,s

s−r∣∣ f(β θ−t θ

+se−tej

θt)− f(β)∣∣ dt ,

and merely the simpler commutation formula (I.16) :

`(θ−t θ

+se−t ej

θt)

= `(θ+se−2t ej

)= O(se−2t) = O(s), for 2 ≤ j ≤ d .

We have the following general statement, derived from Ito’s Formula.


Proposition VIII.6.2 Consider a closed longitudinal 1-form Ω havingcontinuous Lk-derivatives (for k ∈ 0, 2, . . . , d). Then for any t ≥ 0we almost surely have :∫

ξ[0,t]

Ω = MΩt + 1

2

∫ t

0

div Ω (ξs) ds ,

with div Ω := L0(Ω(E1)) +d∑j=2

Lj(Ω(Ej)

)+ (1− d) Ω(E1),

and a continuous martingale MΩt having quadratic variation :

〈MΩ〉t =

∫ t

0

(Ω(E1)

2 +d∑j=2

Ω(Ej)2)

(ξs) ds .

Proof Set e := (e0, . . . , ed) ∈ Fd and F (β) :=

∫ β

e

Ω . Then F is a C2

function on Fd, and Ito’s Formula (VII.3) can be applied to the functionZs 7→ F (ξs) = F (ξ0 Zs) and to Equation (VII.11), yielding :∫ξ[0,t]

Ω = F (ξt)− F (ξ0) =

∫ t

0

∑j=0,2,...,d

LjF (ξs) dWjs + 1

2

∫ t

0

DF (ξs) ds .

Now writing Ω = Ω0E∗1 +d∑j=2

ΩjE∗j and using Formula (VIII.7), we have

L0F (ξ) =dodε

∫ ξ θε

ξ

Ω =dodε

∫ ε

0

Ω0(ξ θs) ds = Ω0(ξ) = Ω(E1)(ξ)

for any ξ ∈ Fd, and similarly LjF = Ωj = Ω(Ej) for 2 ≤ j ≤ d . Hence,we get the claimed expression for 〈MΩ〉t , and for div Ω = DF as well,using the expression (III.3) of D.


Let us apply Proposition VIII.6.2 to ωf (of Formula (VIII.9) in Defini-tion VIII.4.2.1). This is licit by Lemmas VIII.4.2.2 and VIII.6.1. Notic-ing moreover that Ljfj = −U2L2

jf , by Formula (VIII.9) and

Lemma VIII.4.2.2, we obtain∫ξ[0,t]

ωf = M ft + 1

2

∫ t

0

Kf(ξs) ds (VIII.10)

at once, with

Kf := L0f −d∑j=2

U2L2jf + (1− d)f , (VIII.11)

M ft being a continuous martingale having quadratic variation given by :

〈M f〉t =

∫ t

0

(f 2 +

d∑j=2

f 2j

)(ξs) ds . (VIII.12)

Lemma VIII.6.3 The function Kf of Formula (VIII.11) is Γ-inva-riant, SO(d−1)-invariant, bounded

(which we can summarize by : Kf ∈

L∞(M, µ)), and Holderian.

Proof We already observed in Lemma VIII.4.2.2 that the fj and theLjfj are bounded. L0f is SO(d − 1)-invariant since f is and sincethe geodesic flow commutes with the SO(d−1)-action on Fd. We must

then show thatd∑j=2

U2L2jf is SO(d− 1)-invariant. The commutation of

the geodesic flow with the SO(d − 1)-action shows that it is sufficient

to verify the SO(d − 1)-invariance ofd∑j=2

L2jf . Now, considering any


% ∈ SO(d − 1), and using the SO(d − 1)-invariance of f and LemmaI.4.2, we have :

d∑j=2

L2jf (β %) =

d∑j=2

d2o

dt2f(β % θ+

tej)

=d∑j=2

d2o

dt2f(β % etEj%−1

)=

d∑j=2

d2o

dt2f(β θ+

t%(ej)

).

In particular, if % is a rotation of angle α in the plane ek, el, we have :

d∑j=2

L2jf (β %) =

d2o

dt2

[∑j 6=k,l

f(β θ+

tej

)+ f(β θ+

t(cosα)ek−t(sinα)el

)+ f(β θ+

t(sinα)ek+t(cosα)el

)]=

[∑j 6=k,lL2jf +

[(cosα)Lk − (sinα)Ll

]2f +

[(sinα)Lk + (cosα)Ll

]2f

](β)

=d∑j=2

L2jf (β).

As any % ∈ SO(d− 1) is a finite product of planar rotations, this provesthe wanted invariance.

It remains to show that Kf is Holderian. By Formula (VIII.10), thisamounts to verify that the U2L2

jf are Holderian, which is provided byLemma VIII.6.1 (whose proof can be applied directly to the Holderianfunction L2

jf).

Proposition VIII.6.4 We have

∫Kf dλΓ = 0

(K was defined in

Formula (VIII.11)).


Proof By Formula (VIII.8) defining U2, for 2 ≤ j ≤ d , β ∈ Fd andS > 0 we have :

U2L2jf(β) = O

(e−2S

)+

∫ S

0

e−2sL2jf(β θs

)ds , and

e−2sL2jf(β θs

)= e−2s do

dtLjf

(β θs θ

+t ej

)= e−2s do

dtLjf

(β θ+

est ejθs)

= e−sdodtLjf

(β θ+

t ejθs)

= Lj(e−sLjf(· θs)

)(β) = L2

j

(f(· θs)

)(β) .

Hence,

d∑j=2

U2L2jf = O

(e−2S

)+

∫ S

0

(D − L2

0 + (d− 1)L0

)f(· θs)ds .

Now, ∫ S

0

L0f(· θs)ds =

∫ S

0

d

dsf(· θs)ds = f

(· θS

)− f ,

and

∫ S

0

L20f(· θs)ds = L0f

(· θS

)− L0f .

Therefore by the definition (VIII.11) of Kf we obtain :

Kf = −D(∫ S

0

f(· θs) ds)

+ L0f(· θS) + (1− d)f(· θS) +O(e−2S

).

Hence, using the right invariance of λΓ and the hypothesis made on f ,we find : ∫

Kf dλΓ

= −∫D(∫ S

0

f(· θs) ds)dλΓ +

∫L0f dλ

Γ + (1− d)

∫f dλΓ +O

(e−2S

)= −

∫ (∫ S

0

f(· θs) ds)D∗1 dλΓ +O

(e−2S

)= O

(e−2S

),

which gives the result by letting S go to infinity.


VIII.7 Sinai’s Central Limit Theorem

The aim of this section is to complete the proof of the following theo-rem, essentially due to Y. Sinai ([Si1], [Si2]), who treated the cocompactcase. (See also M. Ratner [Ran] in the compact case too, but in non-constant curvature). The cofinite case appeared in [LJ2], and the infinitecase in [EFLJ3].

Theorem VIII.7.1 Let f be a Γ-invariant, SO(d−1)-invariant boun-ded real function on Fd, of class C2 with bounded and Holderian deriva-

tives, such that

∫f dλΓ = 0 . Then for all a ∈ R we have :

limt→∞

∫exp

(a√−1√t

∫ t

0

f(βθs) ds

)λΓ(dβ) = covol(Γ)× exp

(− a2

2 V(f)),

(VIII.13)where

(Kf being given by Formula (VIII.11)

)V(f) :=

[(d−1) covol(Γ)

]−1∫ [

(f+12 L0V Kf)2+

d∑j=2

(fj+12 LjV Kf)2

]dλΓ

(VIII.14)vanishes if and only if f equals L0h , for some h ∈ L2(Fd, λΓ).

Nota Bene The first claim(Formula (VIII.13)

)equivalently reads : for

all real a we have

limt→∞

λΓ

[β ∈ Fd

∣∣∣ ∫ t

0

f(βθs) ds ≤ a√tV(f)

]=

covol(Γ)√2π

∫ a

−∞e−s

2/2 ds .

The kernel V which appears in Theorem VIII.7.1, in the definitionof the variance V , is the potential kernel of the semi-group (Pt) :

V :=

∫ ∞0

Pt dt , (VIII.15)

VIII.7. SINAI’S CENTRAL LIMIT THEOREM 289

which, as we saw in Theorem VIII.3.4, makes sense on centred Γ-inva-riant functions which are regular enough, such as Kf of Section VIII.6(by Lemma VIII.6.3 and Proposition VIII.6.4). By Remark VIII.3.3, Vpreserves SO(d− 1)-invariance too.

To give sense to the expression (VIII.14) in Theorem VIII.7.1, on theone hand we need to justify the existence of L2-derivatives LjV Kf .On the other hand, to prove Theorem VIII.7.1 we want to use Proposi-tion VIII.6.2 and its consequence Formula (VIII.10) (in Section VIII.6),but with

[ωf + 1

2 d(V Kf)]

instead of ωf , in order to get rid of the

problematic part

∫ t

0

Kf(ξs) ds in Formula (VIII.10). Hence we must

express this term in the form of :∫ t

0

Kf(ξs) ds = V Kf(ξt)− V Kf(ξ0) + a martingale term .

To do this, the idea is to show the following, which will apply to F =Kf , thanks to Lemma VIII.6.3 and Proposition VIII.6.4.

Proposition VIII.7.2 Any centred and bounded rotationnally r-Holde-rian function F on M admits a bounded potential V F with L2-deriva-

tives LjV F such that, setting Vb :=

∫ b

0

Pt dt : (for j ∈ 0, 2, . . . , d)

(i)∥∥∥V F − VbF dt∥∥∥

L2(λΓ)≤(‖F‖∞ + |||F |||r

)×O(e−δ b) ;

(ii)∥∥∥LjV F − LjVbF dt∥∥∥

L2(λΓ)≤(‖F‖∞ + |||F |||r

)×O(e−δ b/2).

Proof 1) Assume first that F has continuous bounded derivativesLjF, L2

jF , and set

E(F ) :=

∫ ∑j=0, 2≤j≤d

|LjF |2 dλΓ = −∫F DF dλΓ, (VIII.16)


the last equality holding since∫ ∑j=0, 2≤j≤d

|LjF |2 dλΓ = −∫ (L2

0F +d∑j=2

L2jF)F dλΓ

= −12

∫(DF +D∗F )F dλΓ.

Observe then that by Lemma VIII.1.2 for any b > 0 we have :

PbF − F =

∫ b

0

d

dtPtF dt = 1

2

∫ b

0

DPtF dt = 12 DVbF . (VIII.17)

Then for any b ≥ c ≥ 0, using Formulas (VIII.16), (VIII.17) and Theo-rem VIII.3.4 we get :∥∥(Vb − Vc)F

∥∥2≤ C

(‖F‖∞ + |||F |||r

)(e−δ c − e−δ b

)/δ ,

and

E((Vb − Vc)F

)=

∫(Vc − Vb)F ×D(Vb − Vc)F dλΓ

= 2

∫(Vc−Vb)F ×(Pb−Pc)F dλΓ ≤ 2

∥∥(Vb−Vc)F∥∥

2

(‖PbF‖2 +‖PcF‖2

)≤ 2C2 δ−1

(‖F‖∞ + |||F |||r

)2 (e−δ b + e−δ c

).

This shows that V F := limb→∞

VbF and LjV F := limb→∞

LjVbF are well

defined in L2(λΓ), and that (i),(ii) hold. Moreover we have(with C ′ :=

(C + 2C2) δ(r,Γ)−1)

:

‖VbF‖2 ≤ C ′(‖F‖∞+|||F |||r

)and

‖V F‖2 = limb→∞‖VbF‖2 ≤ C ′

(‖F‖∞+|||F |||r

), (VIII.18)

andE(V F ) = lim

b→∞E(VbF ) ≤ C ′

(‖F‖∞+|||F |||r

)2. (VIII.19)


By definition of Lj , showing that LjV F defined so is indeed a L2-

derivative amounts to proving that s−1

∫ s

0

θjt LjV F dt goes to LjV F in

L2 as s 0 , where (not to handle heavy expressions) for s ≥ 0, β ∈ Fdwe set :

θ0sF (β) := F (βθs) and θjsF (β) := F (βθ+

s ej) for 1 ≤ j ≤ d .

Note that by Remark VIII.1.3 VbF admits bounded Lj- and L2j-deriva-

tives, since for example

‖LjVbF‖∞ =∥∥∥Lj∫ b

0

PsFds∥∥∥∞

=∥∥∥∫ b

0

LjPsFds∥∥∥∞≤∫ b

0

‖LjPsF‖∞ds <∞.

Now for j ∈ 0, 2, . . . , d and b, s ≥ 0 the regularity of VbF ensures thatwe have :∫ s

0

θjt LjVbF dt = s−1

∫ s

0

d

dt(θjt VbF ) dt = θjs VbF − VbF ,

and then we can let b ∞ , provided the convergences in L2(λΓ)of θjs VbF to θjs V F and of θjsLjVbF to θjsLjV F are uniform in s ∈[0, 1]. Now this is granted by (i) and (ii) proved above for F and by theinvariance of λΓ with respect to θjs

(recall Proposition V.1.1(iv)

).

2) For generic F as in the statement, as is easily seen using a con-volution and a partition of unity we can find a sequence (Fn) of centredand bounded functions on M having continuous bounded derivativesLjF, L2

jF , such that

‖F − Fn‖L2(λΓ) −→ 0 and ‖Fn‖∞ ≤ 2 ‖F‖∞ , |||Fn|||r ≤ 2 |||F |||r .

Then on the one hand, using Remark VIII.3.3 and Theorem VIII.3.4 wehave : ∥∥V F − V Fn∥∥L2(λΓ)

≤∫ ∞

0

∥∥Pt(F − Fn)∥∥2dt


≤∫ ∞

0

min‖F − Fn‖2 , 3C

(‖F‖∞ + |||F |||r

)e−δt

dt

= 2C(‖F‖∞+|||F |||r

)δ−1‖F−Fn‖2×log

[3eC

(‖F‖∞+|||F |||r

)‖F−Fn‖−1

2

]which goes to 0, and similarly for

∥∥VbF−VbFn∥∥2(uniformly with respect

to b), so that (i) holds.

On the other hand, in a way similar to 1) above, for b, n, p ∈ N∗ wehave :

E(Vb(Fn − Fp)

)=

∫Vb(Fp − Fn)DVb(Fn − Fp) dλΓ

= 2

∫Vb(Fp − Fn)

[Pb(Fn − Fp)− (Fn − Fp)

]dλΓ

≤ 2(‖VbFp‖2 + ‖VbFn‖2

)(‖Pb(Fn − Fp)‖2 + ‖Fn − Fp‖2

)≤ 4

(‖VbFp‖2 + ‖VbFn‖2

)‖Fn − Fp‖2 ,

and then using (VIII.18), via b∞ we obtain :

E(V (Fn − Fp)

)≤ 8C ′

(‖F‖∞ + |||F |||r

)× ‖Fn − Fp‖2 .

This shows that (for j ∈ 0, 2, . . . , d) (LjV Fn) is a Cauchy sequence inL2(λΓ), so that it converges to a limit we denote by LjV F . Moreoverfor n, b ∈ N∗ we have :

E(V (Fn − F )

)≤ 8C ′

(‖F‖∞ + |||F |||r

)× ‖Fn − F‖2 , (VIII.20)

andE(Vb(Fn − F )

)≤ 8C ′

(‖F‖∞ + |||F |||r

)× ‖Fn − F‖2 ,

which entail readily that (ii) holds.

We are now left with the verification that LjV F is indeed the L2-derivative at 0 of θjsV F . As in 1) above, since we know that this holdswith Fn instead of F , it is enough to make sure that the convergences inL2(λΓ) of θjs V Fn to θjs V F and of θjsLjV Fn to θjsLjV F are uniform


with respect to s ∈ [0, 1]. Now this is granted by the above L2(λΓ)-estimates and again by the invariance of λΓ with respect to θjs .

As already quoted, Lemma VIII.6.3 and Proposition VIII.6.4 allowto apply Proposition VIII.7.2 to the function Kf = divωf of Formula(VIII.10), which gives full sense to the expression (VIII.14) in TheoremVIII.7.1. This allows also to apply Ito’s Formula to V Kf , or alterna-tively, to apply Formula (VIII.10) of Section VIII.6 with[ωf + 1

2 d(V Kf)]

instead of ωf . This yields the following.

Proposition VIII.7.3 For any t ≥ 0 we have :∫ξ[0,t]

ωf = 12 V Kf(ξ0)− 1

2 V Kf(ξt) +Mft ,

where(M

ft

)is a continuous martingale having quadratic variation :

⟨M

f ⟩t

=

∫ t

0

[(f + 1

2 L0V Kf)2 +d∑j=2

(fj + 12 LjV Kf)2

](ξs) ds .

Proof As the existence of div[d(V Kf)

]is not clear, let us use again

an approaching sequence (Fn) for F ≡ Kf as in the proof of PropositionVIII.7.2, and apply Proposition VIII.6.2 and Formula (VIII.10), to[ωf + 1

2 d(VbFn)]

instead of ωf or[ωf + 1

2 d(V Kf)].

By Proposition VIII.6.2 and Formula (VIII.17) we have :

div[d(VbFn)

]= D(VbFn) = 2PbFn − 2Fn ,

so that we get in this way :∫ξ[0,t]

ωf + 12 VbFn(ξt)− 1

2 VbFn(ξ0) = M f,b,nt + 1

2

∫ t

0

[Kf+Pb Fn−Fn](ξs) ds ,(VIII.21)


where by Formula (VIII.12) the continuous martingale(M f,b,n

t

)has qua-

dratic variation :⟨M f,b,n

⟩t

=

∫ t

0

[(f + 1

2 L0VbFn)2

+d∑j=2

(fj + 1

2 LjVbFn)2](ξs) ds .

(VIII.22)The result follows from Proposition VIII.7.2 (including (VIII.20)), lettingb, n go to infinity.

So far, it remains to prove a Central Limit Theorem for the martingale(M

ft

). Recall from Definition VIII.2.1 and Proposition VIII.2.2 that the

diffusion (ξs) is stationary with semi-group (Pt) and law µ(proportional

to λΓ)

on Γ\Fd.

Lemma VIII.7.4 Let (Mt) be a continuous real martingale of the form

Mt =∑

j=0,2,...,d

∫ t

0

ψj(ξs) dWjs , for some ψj ∈ L2(M, µ). Then the law

of Mt/√t converges, as t→∞ , towards the centred Gaussian law with

variance

∫Γ\Fd

H2 dµ , where H2 :=∑

j=0,2,...,d

ψ2j .

Proof 1) By Proposition V.3.2, the ergodic theorem V.2.3 applies to(ξt), which goes indeed to infinity in PSO(1, d) by Formulas VIII.3 and

Proposition VI.3.3. This shows that 〈M〉t/t goes to

∫Γ\Fd

H2 dµ almost

surely as t→∞ .

2) Suppose here that H2 ∈ L∞(M, µ), and note that for any real a

Nt := exp

(a√−1√t

Mt +a2

2t〈M〉t

)defines a local martingale, which is

bounded since its modulus is plainly dominated by ea2‖H2‖∞/2. Hence it

is a bounded martingale. Then taking advantage of the non-negativity


of H2 and setting F := H2 −∫H2dµ , we have :∣∣∣∣E[Nt exp

(− a2

2t

∫ t

0

F (ξs) ds)]− 1

∣∣∣∣=

∣∣∣∣E[Nt

[exp(− a2

2t

∫ t

0

F (ξs) ds)− 1]]∣∣∣∣

≤ ‖Nt‖2 ×∥∥∥ exp

(− a2

2t

∫ t

0

F (ξs) ds)− 1∥∥∥

2

≤ ea2

2 ‖H2‖∞ × ea2

2 ‖H2‖1 a2

2

∥∥∥ t−1

∫ t

0

F (ξs) ds∥∥∥

2,

which goes to zero by 1) above. Therefore we finally obtain the conver-gence of

E[exp

(a√−1

Mt√t

)]= E

[Nt exp

(− a2

2t

∫ t

0

F (ξs) ds)]× exp

(− a2

2

∫H2 dµ

)towards exp

(− a2

2

∫H2 dµ

), which concludes the proof in the case of

a bounded integrand H2.

3) To complete the proof, we reduce the general case of H2 ∈L1(M, µ), i.e., of ψj ∈ L2(M, µ) to the above one. For any n ∈ N∗, set

ψnj := minn,max−n, ψj

, and Mn

t :=

∫ t

0

∑j=0,2,...,d

ψnj (ξs) dWjs .

Then∥∥∥Mt−Mn

t√t

∥∥∥2

2=

∑j=0,2,...,d

∥∥ψj − ψnj ∥∥2

2goes to 0 as n→∞ , and there-

fore

lim supn→∞

∣∣∣∣E[exp

(a√−1

Mt√t

)− exp

(a√−1

Mnt√t

)] ∣∣∣∣2


≤ lim supn→∞

E

[∣∣∣∣ exp

(a√−1

Mt −Mnt√

t

)− 1

∣∣∣∣2]

= 4 lim supn→∞

E[sin2

(aMt −Mn

t

2√t

)]= 0 ,

uniformly with respect to t . Hence we can apply 2) above to each (Mnt )

and go to the limit :

limt→∞

E[exp

(a√−1

Mt√t

)]= lim

n→∞limt→∞

E[exp

(a√−1

Mnt√t

)]

= limn→∞

exp(− a2

2

∫ ∑j=0,2,...,d

(ψnj )2 dµ)

= exp(− a2

2

∫H2 dµ

).

Lemma VIII.7.5 As t → ∞ , the law of t−1/2

∫ξ[0,t]

ωf converges

towards the centred Gaussian law with variance(recall from Definition

VIII.2.1 that µ := covol(Γ)−1λΓ)

:

V0(f) :=

∫ [(f + 1

2 L0V Kf)2 +d∑j=2

(fj + 12 LjV Kf)2

]dµ .

Proof 1) We can apply Lemma VIII.7.4 to the martingale(M

ft

)appearing in Proposition VIII.7.3. Indeed, on the one hand as al-ready quoted, Lemmas VIII.4.2.2, VIII.6.3 and Propositions VIII.6.4,VIII.7.2 show that H2 ∈ L1(M, µ). And on the other hand, the contin-

uous martingale(M

ft

)of Proposition VIII.7.3 is indeed of the Brownian

form considered in Lemma VIII.7.4, since (by the proof of Proposition

VIII.7.3) it reads actually Mft = M f

t + 12

∑j

∫ t

0

LjV Kf(ξs) dWjs , with

VIII.7. SINAI’S CENTRAL LIMIT THEOREM 297(M f

t

)of the wanted form, by (VIII.10), (VIII.9) and the proof of Propo-

sition VIII.6.2.

2) Then, t−1/2(V Kf(ξ0)− V Kf(ξt)

)goes to zero in probability as

t → ∞ , since V Kf(ξt) is stationary (recall Definition VIII.2.1 andProposition VIII.2.2). Hence, applying Proposition VIII.7.3, we see

that we have to deal with limt→∞

E[

exp(a√−1 M

ft /√t)]

.

Now, by 1) above we have :

limt→∞

E[

exp(a√−1 M

ft /√t)]

= exp(− a2

2V0(f)

).

Finally, Theorem VIII.5.1 and the previous lemma VIII.7.5 imply atonce that :

limt→∞

∫exp

(a√−1√t

∫ (d−1)t/2

0

f(β θs) ds

)dµ(β) = exp

(− a2 V0(f)/2

).

Recalling that (in Definition VIII.2.1) µ = covol(Γ)−1λΓ, and changinga into a/

√d− 1 , which changes V0(f) into V(f) of Formula (VIII.14),

we deduce immediately the first claim of Sinai’s Central Limit TheoremVIII.7.1.

The last assertion of Theorem VIII.7.1 is easy : it is indeed clearfrom Formula (VIII.14) that if V(f) = 0 , then f = L0(−1

2 V Kf) λΓ-

almost everywhere, and V Kf belongs indeed to L2(λΓ) by PropositionVIII.7.2 ; and reciprocally, if f = L0h , then by Formula (VIII.1)

t−1/2

∫ t

0

f(βθs) ds = t−1/2[h(βθt)− h(β)

]goes to zero in µ-probability, by the invariance of Proposition (V.1.1,iv). This forces clearly V(f) = 0 .

The proof of Theorem VIII.7.1 is now complete.


Remark VIII.7.6 To establish Theorem VIII.7.1, we used actuallyonly the following slightly weaker assumption :

f and L0f bounded, rotationally Holderian on Hd× ∂Hd, and continu-ous along the stable leaves, and, for 2 ≤ j ≤ d , Ljf and L2

jf bounded

and Holderian on Fd.

Remark VIII.7.7 The same argument applied to a function f ver-ifying the same assumptions, except that its average is not necessarilyzero, yields the convergence in probability of the ergodic mean towardsits average. This is enough to establish the above Central Limit The-orem (recall that the ergodicity was used in the contour deformationargument, precisely in the proof of Theorem VIII.5.1). But this result isclearly weaker than the mixing theorem established in Section V.3.

VIII.8 Notes and comments

The central limit theorem for the geodesic flow appeared originally inthe articles of Sinai [Si1], [Si2] and Ratner [Ran] in the cocompact case.The stochastic analysis method we have used in this chapter originatesfrom the article [LJ2], and differs from the standard method, based onMarkovian partitions and coding.

Other noteworthy articles on that subject are [L2], [L3], [CL].

This type of study was then extended to related questions :

- central limit theorem for the geodesic flow in hyperbolic manifolds ofinfinite volume : [EFLJ3] ;

- singular windings of geodesics (in particular, near cusps in manifoldsof constant negative curvature), in finite volume : [LJ1], [GLJ], [LJ3],[F1], [ELJ], [F2], [F3], [EFLJ4] ; and in infinite volume : [EFLJ1], [EF] ;

- exit law [EFLJ2], and counting closed geodesics [BP1], [BP2].

Chapter IX

Appendix relating to geometry

IX.1 Structure of pseudo-symmetric matrices

We describe here the reduction of symmetric endomorphisms in Minkowski’s space R1,d.

This is the Lorentzian counterpart of a well known result in the Euclidean framework.

We considered in Section I.1.5 the diagonal matrix J = diag(1,−1, . . . ,−1) ∈ M(d+ 1)having diagonal entries (1,−1, . . . ,−1), matrix of the Minkowski pseudo-metric 〈·, ·〉 in R1,d.In other words, Jij = J ij = 1i=j=0 − 1i=j 6=0.

Definition IX.1.1 An endomorphism T of R1,d is (pseudo-) symmetric if for any u, v ∈R1,d, 〈Tu, v〉 = 〈u, Tv〉 ; i.e., if the associated bilinear form R1,d × R1,d 3 (u, v) 7→ 〈Tu, v〉is symmetric. Given any pseudo-orthonormal base b = (b0, b1, . . . , bd) of R1,d, we set Tij :=〈Tbi, bj〉, and call ((Tij)) the matrix of T in the base b .

Thus the symmetry of T reads as usual Tij = Tji in terms of its matrix (in any pseudo-

orthonormal base). Note that the j-th coordinate of any u ∈ R1,d is uj := J jj〈bj , u〉, so

that 〈Tu, v〉 =∑

0≤i,j≤dTij u

i vj ; while (Tu)j = J jj〈bj , Tu〉 =d∑i=0

J jj Tji ui.

Our aim here is to establish the following reduction theorem for symmetric endomorphismsof R1,d (stated without proof in [HE], pages 89-90). This is actually a result of simultaneousreduction of such endomorphism and of the Minkowski pseudo-metric. It is more complicatedthan its classical Euclidean analogue, as was already the case with Theorem I.5.1 for thereduction of isometries.

299

300 CHAPTER IX. APPENDIX RELATING TO GEOMETRY

Theorem IX.1.2 ([HE], pages 89-90) Let T be a real symmetric endomorphism of R1,d.Then there exists a pseudo-orthonormal base (b0, . . . , bd) in which the matrix ((Tij)) of Treduces to one of the following types (with real λ, λj , µ) :

(i1) =

λ 0 . . . 00 λ1 . . . 0...

.... . .

...0 0 . . . λd

; (i2) =

ε+ µ ε 0 . . . 0ε ε− µ 0 . . . 00 0 λ2 . . . 0...

......

. . ....

0 0 0 . . . λd

(with ε = ±1) ;

(i3) =

−λ µ 0 . . . 0µ λ 0 . . . 00 0 λ2 . . . 0...

......

. . ....

0 0 0 . . . λd

(with µ 6= 0) ; (ii) =

−λ 0 1 0 . . . 00 λ 1 0 . . . 01 1 λ 0 . . . 00 0 0 λ3 . . . 0...

......

.... . .

...0 0 0 0 . . . λd

.

We begin by quoting two easy facts about R1,d.

Remark IX.1.3 (i) If 〈u, u〉 = 1 and w ∈ u⊥ \ 0, then 〈w,w〉 < 0 .

(ii) If two lightlike vectors u, v are linearly independent, then 〈u, v〉 6= 0 , and u′, v′ :=u± (2〈u, v〉)−1v generate the same plane as u, v and are such that 〈u′, u′〉 = −〈v′, v′〉 = 1and 〈u′, v′〉 = 0 .

The main step of the reduction is the following.

Lemma IX.1.4 Let T be a (real) symmetric endomorphism in Minkowski space R1,d. Thenthere exists a pseudo-orthonormal base (b0, . . . , bd) of R1,d such that one of the two followingmutually exclusive cases occurs.

- Either (i) : b2, . . . , bd are eigenvectors of T , and there exist real α, β, γ such that

Tb0 = αb0 − βb1 and Tb1 = γb1 + βb0 .

- Or (ii) : b3, . . . , bd are eigenvectors of T , and there exist real q 6= 0 and a, λ such that

Tb0 = ab0 − (λ+ a)b1 − q b2 , T b1 = (a+ λ)b0 − (2λ+ a)b1 − q b2 , T b2 = q(b0 − b1)− λ b2 .

IX.1. STRUCTURE OF PSEUDO-SYMMETRIC MATRICES 301

Remark IX.1.5 The two types of symmetric endomorphisms in Lemma IX.1.4 correspondto the following two types of symmetric matrices ((Tij)) (with q 6= 0) :

α β 0 . . . 0β γ 0 . . . 00 0 λ2 . . . 0...

......

. . ....

0 0 0 . . . λd

;

a λ+ a q 0 . . . 0λ+ a 2λ+ a q 0 . . . 0q q λ 0 . . . 00 0 0 λ3 . . . 0...

......

.... . .

...0 0 0 0 . . . λd

.

Proof Necessarily T admits a complex-valued eigenvalue at least : there exist real numbersx, y and real vectors u, v (non vanishing both), such that

T (u+√−1 v) = (x+

√−1 y)(u+

√−1 v), or equivalently Tu = xu−yv and Tv = yu+xv .

Then x〈u, v〉−y〈v, v〉 = 〈Tu, v〉 = 〈u, Tv〉 = y〈u, u〉+x〈u, v〉, whence, either y = 0 and thenu or v is an eigenvector associated to the eigenvalue x , or 〈u, u〉+〈v, v〉 = 0 and u, v span astable plane

(indeed, if y 6= 0 and αu+βv = 0, then 0 = αTu+βTv = (αx+βy)u+(βx−αy)v ,

whence 0 = (αx+ βy)β − (βx− αy)α = (α2 + β2)y and then α = β = 0 .)

In the latter case, if the linearly independent vectors u, v are neither lightlike nor orthogonal :up to a multiplication and an exchange, we can suppose that 〈u, u〉 = 1 = −〈v, v〉 and then,replacing v by (1 + 〈u, v〉−2)−1/2(u− 〈u, v〉−1v), that they are orthogonal.

Again in the case 〈u, u〉+〈v, v〉 = 0 , but if the linearly independent vectors u, v are lightlike,then they cannot be orthogonal by Remark (IX.1.3, (ii)), so that replacing u, v by

u+ v, u− v we are brought back to the preceding case.

So far, we found that either T admits a real eigenvector b 6= 0 (having pseudo-norm 1, 0, or−1), or it admits a real stable plane b0, b1 such that 〈b0, b0〉−1 = 〈b0, b1〉 = 〈b1, b1〉+1 = 0 ,and necessarily T operates in b0, b1 as in the statement.

Up to replacing T by some T + α1 , we can restrict to the case of an invertible T .

Note then that the orthogonal V ⊥ of any subspace V preserved by T (i.e., TV = V ) has to bepreserved by T , and that eigensubspaces associated with distinct eigenvalues are orthogonal.So that we can decompose R1,d = V0 ⊕ . . . ⊕ Vk into a direct and orthogonal sum of stablesubspaces Vj , such that : any Vj contains a real non-zero eigenvector bj , or is a planeb0, b1 as described in the statement ; if bj is a non-lightlike eigenvector, then Vj = Rbjis a line ; at most one Vj can contain a non-spacelike vector (by Remark IX.1.3) ; some Vjmust contain a timelike vector.

In particular, if V0 is a plane b0, b1 as described in the statement, or if each Vj is a line, thenthe decomposition (i) of the statement follows directly. Let us then focus on the remainingcase, of any Vj containing a real non-zero eigenvector bj , Vj being a line for 0 ≤ j < k < d ,


bk being lightlike and not spanning Vk , and such that Tbk = λbk . We can moreover assume,by taking k maximal, that Vk does not contain any non-lightlike eigenvector.

By Remark IX.1.3, R1,d \ Rbk cannot contain any non-spacelike vector orthogonal to bk ,so that we can suppose that 〈bj , bj〉 = −1 for 0 ≤ j < k . Moreover Vk must contain avector vk such that 〈vk, vk〉 = 1 . By Remark (IX.1.3,(i)), we must have 〈bk, vk〉 6= 0 , andthen vk+1 := vk − (〈bk, vk〉)−1bk is such that 〈vk+1, vk+1〉+ 1 = 〈vk, vk+1〉 = 0 , and (up to amultiplication) bk = vk − vk+1 . We can then complete vk, vk+1 into a pseudo-orthonormalbasis vk, vk+1, . . . , vd of Vk, with 〈vi, vi〉 = −1 for k < i ≤ d . Set for k ≤ j ≤ d :

Tvj =d∑i=0

qijvi .

Since T (vk − vk+1) = vk − vk+1 , and by the symmetry of T , we must furthermore have

Tvk = avk+(λ−a)vk+1 +d∑

i=k+2

qivi and Tvk+1 = (a−λ)vk+(2λ−a)vk+1 +d∑

i=k+2

qivi , and

then for k+2 ≤ j ≤ d : Tvj = qj(vk+1−vk)+d∑

i=k+2

qijvi , with qi` = q`i for k+2 ≤ i < ` ≤ d .

By the usual reduction of symmetric real matrices in a Euclidean space, we can assume thatqi` = δi` λi for k + 2 ≤ i ≤ ` ≤ d.

Note then that for any k + 2 ≤ j ≤ d the vector v′j := aj(vk − vk+1) + (λ − λj)vj satisfies

Tv′j := λj v′j and 〈v′j , v′j〉 = −(λ − λj)2, hence is a spacelike eigenvector if λj 6= λ . By the

maximality of k, this forces λj = λ for k + 2 ≤ j ≤ d . Again by the maximality of k, wecan restrict to the case qj 6= 0 for k + 2 ≤ j ≤ d .

Then λ is the only eigenvalue of the restriction of T to Vk(it is immediately seen that

det[(T −µ1)|Vk

]= −(λ−µ)d−k+1, by subtracting the fist column to the second one, and by

adding the first row to the second one).

Furthermore, for k < d − 2, then the vector (vk − vk+1 + qk+3 vk+2 − qk+2 vk+3) would bea spacelike eigenvector. Hence we must have k ≥ d − 2. Since k = d − 1 would only yielda subcase of case (i), we can suppose that k = d − 2 (i.e., Vk has three dimensions). Then(vk − vk+1) is the only eigenvector of the restriction of T to Vk . Note that this case is notreducible to case (i), since it would then have at least to non-collinear (complex-valued)eigenvectors. Finally, re-ordering indexes and changing (λ, q) into (−λ,−q), we get the form(ii) of the statement.

End of the proof of Theorem IX.1.2 A change of pseudo-orthonormal base is given by a ma-trix θ ∈ PSO(1, d), and corresponds to changing the symmetric matrix ((Tij)) into θ ((Tij)) θ

−1.Indeed, if θb′j = bj , then T ′ij = 〈Tb′i, b′j〉 = 〈Tθ−1bi, θ

−1bj〉 = 〈θ T θ−1bi, bj〉.

By Remark IX.1.5, it is clearly enough to consider the cases ((Tij)) =

(α ββ γ

)and ((Tij)) =

IX.1. STRUCTURE OF PSEUDO-SYMMETRIC MATRICES 303

a λ+ a qλ+ a 2λ+ a qq q λ

. In the first case, using θ =

(ch t sh tsh t ch t

)∈ PSO(1, 1), with

th (2t) = −2βα+γ if |α + γ| > 2|β| , we get the diagonal form (i1). If |α + γ| = 2|β| 6= 0 , with

e−2t = |β| (and −ch t instead of ch t for (α + γ)β < 0), we get the form (i2). Finally if

|α+ γ| < 2|β| , with e4t = 2β−α−γ2β+α+γ and moreover µ = β

√1− (α+γ

2β )2 and λ = γ−α2

, we get

the form (i3).

In the second case, using first θ =

ε ch t ε sh t 0ε sh t ε ch t 0

0 0 1

∈ PSO(1, 2) with t := − log |q|

and ε being the sign of q , we get the reduction of ((Tij)) to

u− λ u 1u u+ λ 11 1 λ

, with

u := (a + λ)q−2. Using then θ′ =

1 + u2/8 −u2/8 −u/2u2/8 1− u2/8 −u/2−u/2 u/2 1

∈ PSO(1, 2), we get the

last form (ii) of the statement.

As in ([H-E], 4.3), the following is easily deduced from Theorem IX.1.2.

Remark IX.1.6 ([HE], 4.3) Let T be a real symmetric endomorphism of R1,d which satis-fies the so-called “Weak Energy Condition” : 〈Tζ , ζ〉 ≥ 0 for any timelike vector ζ . Then insome pseudo-orthonormal base, its matrix ((Tij)) is of one of the following types of TheoremIX.1.2 (with λ, λj , µ real) :

- either ((Tij)) is of type (i1) with λ ≥ 0 and λ+ λi ≥ 0 for 1 ≤ i ≤ d ;

- or ((Tij)) is of type (i2) with ε = 1 , µ ≥ 0 , and λi ≥ 0 for 2 ≤ i ≤ d .

The “Dominant Energy Condition” requires that moreover Tζ be non-spacelike. This hap-pens for type (i1) if and only if |λi| ≤ λ for 1 ≤ i ≤ d ; and for type (i2) if and only ifε = 1 and 0 ≤ λi ≤ µ for 2 ≤ i ≤ d .

The following variant of the reduction of symmetric endomorphisms can be also deduced fromLemma IX.1.4 and Remark IX.1.5.

Remark IX.1.7 Any symmetric matrix ((Tij)) can be decomposed as a linear combination

of (at most d+ 1) rank 1 matrices : Tij =r∑

n=0qn Ui(n)Uj(n), with 0 ≤ r ≤ d . Moreover,

it is always possible that at most U(0) be non-spacelike in this decomposition.


IX.2 Full commutation relation in PSO(1, d)

We have already seen the fundamental commutation relations (I.16) between the matricesθt and θ+

x of PSO(1, d) (recall Proposition I.4.3), and between these and a rotation matrix% ∈ SO(d) (recall Theorem II.6.1). We did not need the full expression of the relation inthe latter case, so that the description of Theorem II.6.1 was not complete. Here we specifythis commutation relation completely. Since r ∈ SO(d− 1) commutes with matrices θt andsatisfies : r θ+

x r−1 = θ+

rx for any x ∈ Rd−1 (by Proposition I.4.3), it is sufficient to considerplanar rotation matrices (with α 6= 0) Rα,σ introduced in Section II.6 (after Theorem II.6.1),and also considered in Proposition III.3.2.

As the general commutation relation between Rα,σ and any θ+x θlog y is somewhat heavy,

it happens to be more tractable to split such relation into three cases, namely when x = 0 ,when y = 1 and x is collinear to σ , and when y = 1 and x is orthogonal to σ . Thecorresponding three commutation relations are as follows.

Proposition IX.2.1 Fix any σ ∈ Sd−2 and non zero α ∈ R/2πZ, determining a genericplanar rotation Rα,σ (using the notation of Section II.6). Then we have the following three

commutation relations, determining how Rα,σ commutes with the elements of Ad.

(i) For any y > 0 , we have :

Rα,σ θlog y = θ+x′ θlog y′ Rα′,σ , (IX.1)

where

y/y′ = y2 sin2(α2 ) + cos2(α2 ) , x′ =(1− (y′/y)

)cotg (α2 )σ , and cotg (α

′

2 ) = y−1 cotg (α2 ) .

(ii) For any real t , setting x :=(t− cotg (α2 )

)σ , we have :

Rα,σ θ+x = θ+

x′ θlog y′ Rα′,σ , (IX.2)

where

y′ = (1 + t2)−1 sin−2(α2 ) , x′ =(cotg (α2 )− t y′

)σ , and cotg (α

′

2 ) = t .

(iii) For any vector x ∈ Rd−1 which is orthogonal to σ , we have :

Rα,σ θ+x r

ασ,x = θ+

x′ θlog y′ Rα′,σ′ , (IX.3)

where1/y′ = |x|2 sin2(α2 ) + 1 , x′ = (1− y′) cotg (α2 )σ + y′ x ,

Rα′,σ′ is determined by the relation : cotg (α′

2 )σ′ = cotg (α2 )σ − x ,

FULL COMMUTATION RELATION IN PSO(1, d) 305

and rασ,x ∈ SO(d− 1) is the rotation in the plane (σ, x) defined by :

rασ,x σ =2 cotg (α2 )x+ (cotg 2(α2 )− |x|2)σ

|x|2 + cotg 2(α2 ).

Remark IX.2.2 The three commutation relations of Proposition IX.2.1 can be gatheredinto a unique commutation relation, as follows.

For any x ∈ Rd−1, y > 0 , σ ∈ Sd−2 and non-zero α ∈ R/2πZ, we have :

Rα,σ θ+x θlog y r

ασ,x = θ+

x′ θlog y′ Rα′,σ′ , (IX.4)

wherey/y′ = (y2 + |x|2) sin2(α2 ) + cos2(α2 )− 〈x, σ〉 sinα ,

x′ =(1− (y′/y)

)cotg (α2 )σ + (y′/y)(x+ 2〈x, σ〉σ) ,

Rα′,σ′ is determined by the relation : cotg (α′

2 )σ′ =[cotg (α2 )σ − x− 2〈x, σ〉σ

]/y ,

and rασ,x ∈ SO(d− 1) is the rotation in the plane (σ, x) defined by :

rασ,x σ =2(cotg (α2 )− 〈x, σ〉

)x+

(cotg 2(α2 )− |x|2

)σ

|x|2 − 2〈x, σ〉 cotg (α2 ) + cotg 2(α2 ).

In the degenerate case : x = −cotg (α2 )σ , we have : x′ = −x , α′ = π , σ′ = σ , rασ,x = 1 .

In any case we have : cotg (α′

2 )σ′ =(cotg (α2 )σ − x′

)/y′ , sin2(α

′

2 ) = y′y sin2(α2 ), and[(y′)2 + |x′|2 + 1

]/y′ =

[y2 + |x|2 + 1

]/y .

Proof (i) Let us fix a direct pseudo-orthonormal base of M : B := (e0, e1, σ, v3, . . . , vd).We have to verify that Relation (IX.1) holds true on each vector of B . As we are dealingwith elements of PSO(1, d), it is in fact sufficient to consider only d vectors of B .

For 3 ≤ j ≤ d , we first have θ+x′vj = vj − 〈x′, vj〉(e0 + e1) = vj , whence :

Rα,σ θlog y vj = vj = θ+x′ θlog y′ Rα′,σ vj .

Then :

θ+x′ θlog y′ Rα′,σ′ e0 =

[(y′)2 + |x′|2 + 1

2y′

]e0 +

[(y′)2 + |x′|2 − 1

2y′

]e1 +

x′

y′=

[y4 sin2(α2 )+ cos2(α2 )+ y2

2y [y2 sin2(α2 ) + cos2(α2 )]

]e0 +

[y4 sin2(α2 )− cos2(α2 )+ y2 cosα

2y [y2 sin2(α2 ) + cos2(α2 )]

][cosα]e1 +

[1

y′− 1

y

]cotg(α2 )σ


=y2 + 1

2ye0 +

y2 − 1

2y[cosα] e1 +

y2 − 1

2y[sinα]σ = Rα,σ θlog y e0 .

Andθ+x′ θlog y′ Rα′,σ σ = θ+

x′ θlog y′ [cosα′ σ − sinα′ e1]

= cosα′[σ − 〈x′, σ〉(e0 + e1)]− sinα′([

(y′)2 − |x′|2 − 1

2y′

]e0 +

[(y′)2 − |x′|2 + 1

2y′

]e1 −

x′

y′

)

=[cosα′+

y2 − 1

2ysinα sinα′

]σ−

[〈x′, σ〉 cosα′+sinα′

[(y′)2 − |x′|2 − 1

2y′

]](e0 +e1)− sinα′

y′e1

=

[cotg 2(α2 )− y2 + 2(y2 − 1) cos2(α2 )

y2 + cotg 2(α2 )

]σ − [sinα] e1 = [cosα]σ − [sinα] e1 = Rα,σ θlog y σ ,

since the coefficient of −(e0 + e1), equal to[y′

y

]2[sinα

2

][y4 sin2(α2 )−y2+cos2(α2 )+y2−(1−y2)2 cos2(α2 ) sin2(α2 )−[y2 sin2(α2 )+cos2(α2 )]2

],

vanishes.

(ii) We proceed in the same way, to verify that Relation (IX.2) holds true. Firstly,Rα,σ θ+

x and θ+x′ θlog y′ Rα′,σ act identically on v3, . . . , vd . Then

(y′)2 + |x′|2 = y′[(1 + t2) cos2(α2 )− 2t cotg (α2 ) + 1 + cotg 2(α2 )

]implies

(y′)2 + |x′|2 + 1 = y′[2 + (cotg (α2 )− t)2

]= 2y′

[1 + 1

2 |x|2]

and

(y′)2 + |x′|2 − 1

y′= (cotg (α2 )−t)

[cotg (α2 )+sinα−t cosα

]=[|x|2 cosα+2(cotg (α2 )−t) sinα

],

while

x′

y′=[(1 + t2) cos(α2 ) sin(α2 )− t

]σ =

[cos(α2 )− t sin(α2 )

][sin(α2 )− t cos(α2 )

]σ

=[

12 |x|2 sinα− (cotg (α2 )− t) cosα

]σ ,

so that

θ+x′ θlog y′ Rα′,σ e0 =

[(y′)2 + |x′|2 + 1

2y′

]e0 +

[(y′)2 + |x′|2 − 1

2y′

]e1 +

x′

y′

=[1 + 1

2 |x|2]e0 +

[12 |x|2 cosα+ (cotg (α2 )− t) sinα

]e1 +

[12 |x|2 sinα− (cotg (α2 )− t) cosα

]σ

FULL COMMUTATION RELATION IN PSO(1, d) 307

=[1 + 1

2 |x|2]e0 + 1

2 |x|2[cosα e1 + sinασ] +[t− cotg (α2 )

][cosασ − sinα e1] = Rα,σ θ+

x e0 .

Finally, we have :θ+x′ θlog y′ Rα′,σ σ = θ+

x′ θlog y′ [cosα′ σ − sinα′ e1]

= cosα′[σ − 〈x′, σ〉(e0 + e1)

]− sinα′

([(y′)2 − |x′|2 − 1

2y′

]e0 +

[(y′)2 − |x′|2 + 1

2y′

]e1 −

x′

y′

)=

(t2 − 1

1 + t2+ t[

sinα− 2t

1 + t2

])σ − 2t sin2(α2 ) e1

−([ty′ − cotg (α2 )

] t2 − 1

1 + t2+

2t

1 + t2

[(y′)2 − |x′|2 − 1

2y′

])(e0 + e1)

= [t sinα− 1]σ − 2t sin2(α2 ) e1 −(t3y′ − cotg (α2 )(t2 − 1)− t

[ |x′|2 + 1

y′

])e0 + e1

1 + t2

= [t sinα− 1]σ − 2t sin2(α2 ) e1 −(

cotg (α2 )(1− t2)− t[1 + t2 − 2t cotg (α2 )

])e0 + e1

1 + t2

= [t sinα− 1]σ − 2t sin2(α2 ) e1 +[t− cotg (α2 )

](e0 + e1)

=[t− cotg (α2 )

]e0 −

[sinα−

[t− cotg (α2 )

]cosα

]e1 +

[cosα+

[t− cotg (α2 )

]sinα

]σ

= cosασ − sinα e1 − 〈x, σ〉(e0 + cosα e1 + sinασ) = Rα,σ θ+x σ .

(iii) Let us fix a direct pseudo-orthonormal basis ofM :B := (e0, e1, σ, u, v4, . . . , vd), withu := x/|x| . We have again to verify that Relation (IX.3) holds true on d vectors of B .

Firstly, Rα,σ θ+x r

ασ,x and θ+

x′ θlog y′ Rα′,σ act identically on v4, . . . , vd . Then we have :[(y′)2 + |x′|2 + 1

]/y′ = y′

[|x|2 + sin−2(α2 )

]− 2 cotg 2(α2 ) + sin−2(α2 )/y′ = |x|2 + 2 ,

so that

θ+x′ θlog y′ Rα′,σ e0 =

[12 |x|2 + 1](e0 + e1)− [|x|2 sin2(α2 ) + 1

]e1 +

[|x|2 sin(α2 ) cos(α2 )σ + x

]=[1 + 1

2 |x|2]e0 + 1

2 |x|2[cosα e1 + sinασ] + x = Rα,σ θ+x r

ασ,x e0 .

Then

θ+x′ θlog y′ Rα′,σ′(e0 + e1) = θ+

x′ θlog y′[(1− cosα′) e0 + cosα′ (e0 + e1) + sinα′ σ′

]=

2 θ+x′ θlog y′ e0

|x|2 + sin−2(α2 )+

[|x|2 + cotg 2(α2 )− 1

|x|2 + sin−2(α2 )

]y′ (e0 + e1) + 2 θ+

x′

[cotg (α2 )σ − x|x|2 + sin−2(α2 )

]

=

[2 + |x|2

|x|2 + sin−2(α2 )

]e0+

[|x|2 cosα

|x|2 + sin−2(α2 )

]e1+2 sin2(α2 )x′+

[|x|2 + cotg 2(α2 )− 1

|x|2 + sin−2(α2 )

]y′(e0+e1)


+

[2 cotg (α2 )

|x|2 + sin−2(α2 )

][σ + (1− y′)cotg (α2 )(e0 + e1)

]− 2x+ 2 y′ |x|2(e0 + e1)

|x|2 + sin−2(α2 )

=

[1 + y′ − 2 cotg 2(α2 )

|x|2 + sin−2(α2 )

]e0 +

[|x|2 cosα

|x|2 + sin−2(α2 )

]e1 −

[|x|2 + cotg 2(α2 ) + 1

|x|2 + sin−2(α2 )

]y′(e0 + e1)

+(1− y′)[sinα]σ +

[2 cotg (α2 )

|x|2 + sin−2(α2 )

][σ + cotg (α2 )(e0 + e1)

]= [1 + y′] e0 +

[|x|2 cosα

|x|2 + sin−2(α2 )

]e1 − y′(e0 + e1) + [sinα]σ +

[2 cotg 2(α2 )

|x|2 + sin−2(α2 )

]e1

= e0 + [cosα] e1 + [sinα]σ = Rα,σ θ+x r

ασ,x(e0 + e1).

So far, we have proved the existence of rασ,x ∈ SO(d−1) such that Formula (IX.3) holds, withthe given expressions for x′, y′, σ′. It remains only to establish the exact expression of rασ,x .We leave this verification to the interested reader, as an exercise below, which completes theproof of Proposition IX.2.1.

Exercise Perform the necessary computations, in order to verify that the expression givenin the statement of Proposition IX.2.1(iii) indeed defines the rotation rασ,x ∈ SO(d − 1)appearing in Formula (IX.3). Check also that Remark IX.2.2 is correct.

IX.3 The d’Alembertian 2 on R1,d

The d’Alembert operator is naturally associated to the Lorentz-Mobius group PSO(1, d),and induces the hyperbolic Laplacian in a natural way. To specify this second point, we

use polar coordinates in the interioro

C of the solid light cone C : any future-oriented vectorξ ∈ R1,d having positive pseudo-norm can be written in a unique way : ξ = r p , with(r, p) ∈ R∗+ × Hd. In these polar coordinates the d’Alembertian 2 splits in a simple way,very similar to the polar splitting of the Euclidean Laplacian. In the following statement,which is the purpose of this section, we introduce the d’Alembertian 2 by its PSO(1, d)-invariance, and we present its polar splitting yielding the hyperbolic Laplacian.

Proposition IX.3.1 (i) There exists a unique (up to a multiplicative constant) secondorder differential operator 2 without zero order term, the so-called d’Alembertian of R1,d

(defined on C2 functions on R1,d), which is PSO(1, d)-invariant, in the sense that it satisfies

THE D’ALEMBERTIAN 2 309

2(F γ) = (2F ) γ for any F ∈ C2(R1,d) and any γ ∈ PSO(1, d). In any Lorentz baseβ ≡ (β0, . . . , βd) ∈ Fd with corresponding coordinates (ξ0, . . . , ξd) it reads :

2 =∂2

(∂ξ0)2−

d∑j=1

∂2

(∂ξj)2. (IX.5)

(ii) Decomposing any ξ ∈o

C according as ξ = r p with (r, p) ∈ R∗+ × Hd, we have thefollowing decomposition of the d’Alembertian 2 :

2 =∂2

∂r2+d

r

∂

∂r− 1

r2∆ , (IX.6)

where the hyperbolic Laplacian ∆ operates on the p-coordinate.

Proof (i) Write in some Lorentz basis 2 =

d∑i,j=0

aij∂2

∂ξi ∂ξj+

d∑j=0

bj∂

∂ξj. Applying a matrix

γ ∈ PSO(1, d) amounts to changing the Lorentz basis, thereby mapping the coordinate system

ξ to another coordinate system ξ′ = γ ξ , so that we have∂

∂ξj= γkj

∂

∂ξ′k. This entails directly

that the invariance of 2 under γ is equivalent to both conditions :d∑

i,j=0aij γ

ik γ

j` = ak`

andd∑j=0

bj γjk = bk , for any 0 ≤ k, ` ≤ d . The only common eigenvector b to the whole

PSO(1, d) is plainly 0. Hence we are left with the symmetric matrix a = ((aij)), whichhas to satisfy tγ a γ = a for all γ ∈ PSO(1, d). Particularizing to % ∈ SO(d), we see thata′ := ((aij))1≤i,j≤d must satisfy t% a′ % = a′ for all rotations, and must therefore be theopposite of the unit matrix in M(d), up to a multiplicative constant. Hence we must have

a =

a0 a1 a2 . . . ada1 −1 0 . . . 0a2 0 −1 . . . 0...

......

. . ....

ad 0 0 . . . −1

. Then particularizing to the boosts θr yields the conditions

a2 = . . . = ad = 0 , and (a0 − 1) sh r + 2a1ch r = 0 = (a0 − 1) ch r + 2a1sh r , whencea0 − 1 = a1 = 0 , which yields precisely that a = J = diag(1,−1, . . . ,−1), or equivalently,that 2 must read as written in Formula (IX.5). Reciprocally, we noticed already in SectionI.1.5 that tγ J γ = J holds for all γ ∈ PSO(1, d), so that 2 satisfies the required invariance.

(ii) Consider ono

C the canonical coordinates (ξ0, . . . , ξd) of R1,d and the alternativecoordinate system : (

r :=√ξ2

0 − ξ21 − · · · − ξ2

d , p1 :=ξ1

r, . . . , pd :=

ξdr

),


in which we have Hd ≡ r = 1 and (p1, . . . , pd) are coordinates on Hd. Performing thischange of coordinates we have :

∂

∂ξ0=ξ0

r

[∂

∂r−

d∑j=1

pjr

∂

∂pj

], and for 1 ≤ j ≤ d :

∂

∂ξj=

1

r

∂

∂pj+

d∑k=1

pj pkr

∂

∂pk− pj

∂

∂r.

Whence

∂2

∂ξ20

=ξ2

0

r2

∂2

∂r2+r2 − ξ2

0

r3

∂

∂r− 2

ξ20

r3

d∑j=1

pj∂2

∂pj∂r+ξ2

0

r4

d∑j,k=1

pjpk∂2

∂pj∂pk− r2 − 3ξ2

0

r4

d∑j=1

pj∂

∂pj,

and for 1 ≤ j ≤ d :∂2

∂ξ2j

=

=1

r2

∂2

∂p2j

+d∑

k,`=1

p2jpkp`

r2

∂2

∂pk∂p`+ 2

d∑k=1

pjpkr2

∂2

∂pj∂pk+p2

j

∂2

∂r2− 2

pjr

∂2

∂pj∂r− 2

d∑k=1

p2jpk

r

∂2

∂pk∂r

+d∑

k=1

pkr2

∂

∂pk− 1

r

∂

∂r+ 2

p2j

r2

d∑k=1

pk∂

∂pk−p2j

r

∂

∂r+ 2

pjr2

∂

∂pj+

d∑k=1

p2j pk

r2

∂

∂pk.

Hence we get :

2 =∂2

∂r2+d

r

∂

∂r− 1

r2∆ , with ∆ :=

d∑j,k=1

(δjk + pj pk)∂2

∂pj∂pk+ d

d∑j=1

pj∂

∂pj.

Consider then the polar coordinates (%, φ) of Hd, as for Formula (III.11) in Proposition III.5.4.We have p = (ch %)e0 + (sh %)φ and for 1 ≤ j ≤ d : pj = (sh %)φj . Proceeding similarlyto the proof of Proposition III.5.4, let us perform the change of variable from (p1, . . . , pd) to(%, φ2, . . . , φd), in order to verify that ∆ = ∆ , as claimed.

We get so :∂

∂pj=

φjch %

∂

∂%+

1

sh %

d∑`=2

(δj` − φj φ`)∂

∂φ`, whence

d∑j=1

pj∂

∂pj= th %

∂

∂%

and

∂2

∂pj∂pk=

φjφkch 2%

∂2

∂%2+

1

ch % sh %

d∑`=2

(δj`φk + δk`φj − 2φjφkφ`

) ∂2

∂φ` ∂r

+1

sh 2%

d∑`,m=2

(δj` − φjφ`)(δkm − φkφm)∂2

∂φ`∂φm− sh %

ch 3%φjφk

∂

∂%− φj

sh 2%

d∑`=2

(δk` − φkφ`)∂

∂φ`

+1

ch % sh %

((1− δ1k)(δjk − φjφk) + δ1k(δ1j − φ2

1)φj/φ1

) ∂∂%

IX.4. CORE-CUSPS DECOMPOSITION 311

− 1

sh 2%

d∑`=2

((1− δ1k)(δjk − φjφk)φ` + (δj` − φjφ`)φk + δ1k(δ1j − φ2

1)φjφ`/φ1

) ∂

∂φ`,

whence

∆ =∂2

∂%2+ (d− 1) coth %

∂

∂%+

1

sh 2%

[d∑

j,k=2

(δjk − φjφk)∂2

∂φj∂φk− (d− 1)

d∑k=2

φk∂

∂φk

],

which is indeed the expression of ∆ in these coordinates, as we saw in the proof of PropositionIII.5.4.

IX.4 Core-cusps decomposition

Here we prove the core-cusps decomposition theorem IX.4.5 in full generality, i.e., for ageometrically finite and cofinite Kleinian group Γ. Recall that it allowed us to establish thePoincare inequality of Theorem V.4.2.5. Actually we shall reduce it to the following resulton discrete subgroups of Euclidean isometries. See for example

([Rac], Section 5.4

).

Theorem IX.4.1 Any discrete subgroup Γ of Euclidean isometries in Rd has a free Abeliannormal subgroup Γ of finite index, containing all translations in Γ, and acting properly as agroup of translations on some affine subspace of Rd whose dimension is the rank of Γ.

We already quoted (in Corollary IV.4.1.6, page 126, and page 135) that a boundary pointwhich is fixed by some parabolic element of a given Kleinian group Γ was called parabolic.The following is also classical.

Definition IX.4.2 A parabolic (sub)group is a non-trivial Kleinian (sub)group fixing a

parabolic point, and no other point in Hd = Hd ∪ ∂Hd.

The group Sη :=h ∈ PSO(1, d)

∣∣h(η) = η

of hyperbolic isometries fixing a given light

ray η ∈ ∂Hd is conjugate to the group of Euclidean similarities of Rd−1. Any Kleinian groupfixing η is conjugate to a group made of Euclidean similarities, acting discontinously (in thesense of Lemma IV.2.1) on Rd−1. Precisely, we have the following.

Proposition IX.4.3 Consider η ∈ ∂Hd, and any reference Lorentz frame β ∈ Fd(η), whichfixes a Poincare model for Hd, by identification of the current point q = π0(β Tx,y) ∈ Hd

with its Poincare coordinates (x, y) ∈ Rd−1 × R∗+ (with respect to β). Then any hyperbolicisometry g ∈ PSO(1, d) fixes η if and only if it can be identified with a Euclidean similarityof the Poincare upper half-space Rd−1 ×R∗+.


Proof Denote by (xj , yj) (1 ≤ j ≤ 2) the Poincare coordinates (with respect to β) of

qj ∈ Hd, and by (x′j , y′j) those of q′j := g(qj). Let us use Proposition II.1.3, and the beginning

of its proof. We have yj = 〈qj , ηβ0〉−1 and y′j = 〈q′j , ηβ0〉−1 =⟨qj , g

−1(ηβ0)⟩−1

.

Suppose that g(η) = η . Then, we have g−1(ηβ0) = ηg−1(β0) =⟨g(β0), ηβ0

⟩ηβ0 , whence

y′j =⟨g(β0), ηβ0

⟩−1yj . Applying then Formula (II.1) to dist (q1, q2) and dist (q′1, q

′2), which

are equal, and replacing y′j as computed above, we get at once

|x′1 − x′2| =⟨g(β0), ηβ0

⟩−1 |x1 − x2| . This proves that g can indeed be identified with a sim-

ilarity, having dilatation coefficient⟨g(β0), ηβ0

⟩−1.

Reciprocally, it is clear from Formula (II.1) that a similarity of Rd−1×R∗+ defines a hyperbolicisometry fixing η .

We already met the notion of cusp in Corollary IV.4.1.6.

Definition IX.4.4 A cusp of a Kleinian group Γ is a conjugation class (in Γ) of a maximalparabolic subgroup of Γ, so that the set of cusps of Γ is in one-to-one correspondence withthe set of Γ-inequivalent parabolic points, and will be identified with it. The rank of a cusp isthe rank of the corresponding parabolic subgroup (that is, of any free Abelian subgroup havingfinite index, according to Theorem IX.4.1).

Given a fundamental polyhedron P of a Kleinian group Γ, each cusp η of Γ is representedby an ideal vertex of P, say v ∈ ∂P ∩ ∂Hd. Since we can suppose that P has only a finitenumber of vertices (recall that we consider only geometrically finite Kleinian groups Γ), it isalways possible to associate to each vertex v an open horoball Bv (based at v), intersectingonly the sides of P incident to v , and in such a way that these different horoballs be pair-wisedisjoint. Then necessarily, if v represents a cusp η , P ∩Bv is a fundamental domain of thestabilizer Γη of η acting on Bv (this should be clear using Proposition IX.4.3). This impliesthat η is represented by a unique ideal vertex of P, which we identify henceforth with η ;accordingly we shall write Bv = Bη .

For any given cusp η , we fix a reference frame β ∈ Fd(η) such that β0 ∈ ∂Hη , andconsider the corresponding Poincare coordinates (x, y). This amounts to choosing a Poincaremodel Rd−1 ×R∗+, such that η ≡ ∞ and ∂Bη ≡ y = 1.We can describe (of course, up to a hyperbolic isometry) the shape of the solid cusp P ∩Bη(recall Section V.4.2, page 160). Indeed, the sides of P incident to η are now made of verticalgeodesics, so that we have simply in this model P ∩ Bη = P×]1,∞[ , where P := P ∩ ∂Bηis the bottom of the solid cusp P ∩Bη . Moreover, P has also to be a fundamental domainof the stabilizer Γη of η in Γ, to which we apply now Theorem IX.4.1. Thus, denoting

by k ∈ 1, . . . , d − 1 the rank of η , we get a lattice group Zk (conjugate to a finiteindex subgroup of Γη), which is generated by k independent translations in the horosphere

IX.4. CORE-CUSPS DECOMPOSITION 313

y = 1 ≡ Rd−1. Now, this means precisely that P is a finite quotient of a fundamentaldomain P ′ of Zk, which is a parallelepiped in y = 1. Finally, according to the expressionof the hyperbolic metric in the Poincare half-space model Rd−1 × R∗+ (recall PropositionII.1.3), the solid cusp P ∩Bη has finite volume if and only if P has, and then if and only ifP ′ has, hence if and only if η has full rank k = d− 1 .

Now, it happens that all boundary points belonging to ∂P ∩ ∂Hd either are cusps (recallthat above we identified a cusp with the unique ideal vertex of P representing it), or are notlimit points of (any orbit of) Γ, as is guaranteed by

([Rac], Theorem 12.3.4

).

Then, since the set O(Γ) of ordinary (non-limit) points is open in ∂Hd, it happens that anypoint of P ∩ O(Γ) is Γ-equivalent to another one which belongs to the closure of an opensubset of ∂P ∩∂Hd. This latter corresponds to a funnel (recall Remark IV.4.1.9), responsiblefor an infinite volume. Hence clearly, if Γ is cofinite then Γ\Hd cannot have any funnel, sothat in this case all boundary points belonging to ∂P ∩ ∂Hd are cusps.

Hence, once the finite set of horoballs Bη , associated to the cusps as above, have beenremoved from the fundamental polyhedron P, P reduces to its so-called core, which is arelatively compact subset of Hd ∪O(Γ), and then a compact subset of Hd since Γ is cofinite.Therefore we have the following. See also

([Rac], Th. 12.6.6 and Corollary 4

), and

([B1],

GF3 ⇒ GF1).

Theorem IX.4.5 A convex fundamental polyhedron P of a cofinite and geometrically finiteKleinian group Γ is the disjoint union of a finite number of solid cusps, intersections of Pwith open horoballs based at the cusps bounding P, and of a core, which is a compact subsetof Hd. Furthermore, any solid cusp is the quotient by some finite subgroup of Γ, of a solidcusp isometric to P ′× ]1,∞[ , where P ′ ⊂ Rd−1 is the compact fundamental parallelepiped ofsome lattice isomorphic to Zd−1.

Chapter X

Appendix relating to stochasticcalculus

X.1 A simple construction of real Brownian motion

In order to construct a real Brownian motion simply and thereby justify Definition VI.2.1,let us consider the so-called Haar basis of L2

([0, 1]

):

ϕk,j := 2k/2(

1[(j−1)2−k,(j−1/2)2−k

] − 1[(j−1/2)2−k, j2−k

]), for k ∈ N , 1 ≤ j ≤ 2k.

Denote then by φk,j :=

∫ ·0ϕk,j the primitive of ϕk,j which vanishes at 0. We have thus :

φk,j(t) = 2−k/2 Φ(2kt− j + 1) , with (∀u ∈ R) Φ(u) := 1[0,1/2](u)u+ 1]1/2,1](u) (1− u) .

Note that for fixed k ∈ N, the supports Supp(φk,j) = Supp(ϕk,j) =[(j − 1)2−k, j2−k

]have pair-wise disjoint interiors, for 1 ≤ j ≤ 2k. Hence, the following series convergesuniformly on [0, 1] :

∑k,j

φ2k,j(t) =

∑k∈N

2k∑j=1

φ2k,j(t) ≤

∑k∈N

2−k × 14 = 1/2 .

Let ξ0∪ ξk,j | k ∈ N, 1 ≤ j ≤ 2k be a sequence of independent identically distributedN (0, 1) (i.e., normalised centred Gaussian) random variables, and set :

Bt := ξ0 t+∑k∈N

2k∑j=1

ξk,j φk,j(t) , for any 0 ≤ t ≤ 1 . (X.1)

315

316 CHAPTER X. APPENDIX RELATING TO STOCHASTIC CALCULUS

We verify that this series provides the wanted process.

Proposition X.1.1 (i) Almost surely, the above series (X.1) converges uniformly on [0, 1],defining thereby an almost surely continuous process (Bt)0≤t≤1, vanishing at 0.

(ii) For 0 ≤ s ≤ t ≤ 1, the random variable (Bt − Bs) is Gaussian, centred, with variance(t− s).(iii) For 0 = t0 ≤ t1 ≤ . . . ≤ tn ≤ 1, the increments (Bt1 − Bt0), . . . , (Btn − Btn−1) areindependent (for any n ∈ N∗).

Proof (i) Set βk := sup0≤t≤1

∣∣∣∣ 2k∑j=1

ξk,j φk,j(t)

∣∣∣∣ ≤ sup1≤j≤2k

|ξk,j | × 2−k/2−1 . We have :

P[β2k > 2−k log 2k

]≤

2k∑j=1

P[|ξk,j | >

√4 k log 2

]= 2k+1(2π)−1/2

∫ ∞√

4 k log 2e−x

2/2dx ≤ 2−k ,

whence∑k∈N

βk <∞ almost surely, by Borel-Cantelli Lemma, and then the wanted almost

sure uniform convergence. The almost sure continuity follows at once.

(ii) By (i), the choice of the i.i.d. sequence ξk,j, and the convergence of the series∑k,j

φ2k,j , for any real α we have :

E[e√−1 α (Bt−Bs)

]= lim

N→∞exp

[− α2

2

((t− s)2 +

N∑k=0

2k∑j=1

(φk,j(t)− φk,j(s)

)2)].

Now, since 1 ∪ ϕk,j | k ∈ N, 1 ≤ j ≤ 2k is a complete orthonormal system in L2([0, 1]

),

by Parseval’s Formula we have :

(t−s)2+∑k∈N

2k∑j=1

(φk,j(t)−φk,j(s)

)2=

[∫ 1

01[s,t]

]2

+∑k∈N

2k∑j=1

[∫ 1

01[s,t]ϕk,j

]2

=

∫ 1

01[s,t] = t−s .

Hence we get E[e√−1 α (Bt−Bs)

]= e−α

2(t−s)/2 , as wanted.

(iii) As for (ii), setting for any real α1, . . . , αn : h :=

n∑`=1

α` 1[t`−1,t`] , we have :

n∑`=1

α` (Bt` −Bt`−1) = ξ0

∫ 1

0h+

∑k∈N

2k∑j=1

ξk,j

∫ 1

0hφk,j ,

X.1. A SIMPLE CONSTRUCTION OF REAL BROWNIAN MOTION 317

and then

E[e

√−1

n∑=1α`(Bt`−Bt`−1

)]

= exp

[− 1

2

([∫ 1

0h

]2

+∑k∈N

2k∑j=1

[∫ 1

0hϕk,j

]2)]= e−

∫ 10 h

2/2

= exp

[−

n∑`=1

α2` (t` − t`−1)/2

]=

n∏`=1

E[e√−1 α`(Bt`−Bt`−1

)],

which concludes the proof.

Nota Bene The process (Bt)0≤t≤1 of Formula (X.1) and Proposition X.1.1 is a realisationof the real Brownian motion on [0, 1]. If the series in (X.1) is considered without the additionalterm ξ0 t , the resulting process is the real Brownian bridge on [0, 1].

To complete the construction of the real Brownian motion on the whole R+, we considernow a sequence of independent processes (Bn

t )0≤t≤1 , n∈N as in Proposition X.1.1, which isobtained at once, by using a sequence of independent reduced centred Gaussian variables

ξn0∣∣n ∈ N

∪ξnk,j

∣∣ k, n ∈ N, 1 ≤ j ≤ 2k

.

Define the process (Bt)t≥0 by :

Bt :=∑

0≤n≤t−1

Bn1 +B

[t]t−[t] , where [t] := maxn ∈ N |n ≤ t. (X.2)

It is immediate from Proposition X.1.1 that this defines an almost surely continuous process(Bt)t≥0 , which has independent increments and which is such that the random variable(Bt −Bs) is Gaussian, centred, with variance (t− s), for any 0 ≤ s ≤ t .

This completes the construction.

Remark X.1.2 The same construction as above, using, instead of the Haar basis, anyuniformly bounded complete orthonormal system ϕk | k ∈ N∗ of

L20

([0, 1],R

)=f ∈ L2

([0, 1],R

) ∣∣∣ ∫ 1

0f(t) dt = 0

,

the primitive φk of ϕk which vanishes at 0, 1, and a sequence ξk | k ∈ N of independentN (0, 1) random variables, will yield a standard real Brownian motion on [0, 1] as well, bysetting :

Bt := ξ0 t+∑k∈N∗

ξk φk(t) for all t ∈ [0, 1] .

In particular, taking the trigonometric sequence we get the following Fourier expansion ofthe real Brownian motion on [0, 1] : almost surely

Bt = ξ0 t+1

π√

2

∑k∈N∗

1

k

(ξ2k

(1− cos[2πkt]

)+ ξ2k−1 sin[2πkt]

), for any 0 ≤ t ≤ 1 .


Another similar Fourier expansion is

Bt = ξ0 t+

√2

π

∑k∈N∗

ξksin(πkt)

k, for any 0 ≤ t ≤ 1 .

Exercise Justify this last expression(

use Proposition VI.2.2 and compute∑k≥1

sin(πks) sin(πkt)k2

,

by expanding t 7→ t(1− t)(or t 7→ mins, t − st

)in Fourier series on [0, 1]

).

Another classical way of obtaining the Brownian motion is as a limit in law of a randomwalk, as in the following theorem. See for example

([RY], th. XIII.1.9

),([RW1], TH. I.8.2

),

or also([Bi], Th. 37.8

).

Theorem X.1.2.1 (Donsker) Let Sk = X1 + · · · + Xk be a random walk on Z , withelementary step Xj ∈ L2 and centred with variance σ2. For n ∈ N∗ and t ∈ R+, set

Snt :=1

σ√n

(S[nt] +

(nt− [nt]

)X[nt]+1

).

There exist a real Brownian motion (Bt) and a sequence of processes(Snt), such that :

(i) for each n , Sn and Sn have the same law ;

(ii) for any T ∈ R+ , we almost surely have limn→∞

sup0≤t≤T

∣∣Snt −Bt∣∣ = 0 .

X.2 Chaos expansion

We consider here multiple Ito integrals, which are iterated stochastic Ito integrals withrespect to a Rk-valued standard Brownian motion Wt = (W 1

t , . . . ,Wkt ), of the form :∫ s

0dW j1

s1

[ ∫ s1

0dW j2

s2

[. . .

[ ∫ sn−1

0dW jn

sn Fs(s1, . . . , sn)

]. . .

]](

where 1 ≤ j1, . . . , jn ≤ k , s > 0 , and Fs is anyM(d)-valued deterministic function, square

integrable on

(s1, . . . , sn) ∈ Rn∣∣ 0 < sn < . . . < s1 < s

), which we shall write in the more

pleasant form : ∫0<sn<...<s1<s

Fs(s1, . . . , sn) dW j1s1 . . . dW

jnsn .

X.2. CHAOS EXPANSION 319

Lemma X.2.1 The above multiple Ito integrals are pairwise orthogonal in L2. Precisely,

writing Fs ≡ Fs(s1, . . . , sn), Hs ≡ Hs(s1, . . . , sm), FW js :=

∫0<sn<...<s1<s

Fs dWj1s1 . . . dW

jnsn ,

HW is :=

∫0<sm<...<s1<s

Hs dWi1s1 . . . dW

imsm , and φj,is := FW j

s ×HW is , we have :

E(φj,is)

= 1n=m1

(j1,...,jn)=(i1,...,in) × ∫

0<sn<...<s1<sFsHs ds1 . . . dsn .

Proof Using the Ito isometry identity (VI.2) and setting dW j2s2 . . . dW

jnsn =: dW j2...jn

s2...sn , wehave

E(φj,is)

= E[∫ s

0

(∫0<sn<...<s1

Fs dWj2...jns2...sn

)dW j1

s1 ×∫ s

0

(∫0<sm<...<s1

Hs dWi2...ims2...sm

)dW i1

s1

]

= 1j1=i1

∫ s

0E[∫

0<sn<...<s1

Fs dWj2...jns2...sn ×

∫0<sn<...<s1

Hs dWi2...ims2...sm

]ds1 .

By induction, this leads for n ≤ m to : E(φj,is)

=

1(j1,...,jn)=(i1,...,in)

× ∫0<sn<...<s1<s

E[∫

0<sm<...<sn+1<sn

FsHs dWin+1...imsn+1...sm

]ds1 . . . dsn ,

which vanishes for n < m .

Theorem X.2.2 Set for any s ≥ 0 : Λs := exp

[s

(12

k∑j=1

A2j +A0

)]. Then the solution (Xs)

to the linear S.D.E. (VII.1) is given by the so-called Wiener chaos expansion :

Xs = Λs+∑n≥1

∑1≤j1,...,jn≤k

∫0<sn<...<s1<s

ΛsnAjnΛsn−1−sn . . . Aj2Λs1−s2Aj1Λs−s1dWj1s1 . . . dW

jnsn

(X.3)valid for any s ≥ 0 . This series converges in L2-norm, uniformly for bounded s , and allmultiple Ito integrals which constitute it are pair-wise orthogonal in L2.

Proof Set WAt :=

k∑j=1

AjWjt , B :=

(12

k∑j=1

A2j + A0

), and consider Mt := Xt Λs−t , for

fixed s > 0 . Then by Ito’s formula and Equation (VII.1) we have :

dMt = −XtB Λs−t dt+Xt (dWAt +B dt) Λs−t = Xt dW

At Λs−t ,


so that

(∗) Xs − Λs = Ms −M0 =

∫ s

0dMs1 =

∫0<s1<s

Xs1 dWAs1 Λs−s1 .

Iterating this, by induction we obtain :

Xs − Λs =

N−1∑n=1

J An (s) + J AN (s)X for any N ≥ 1 ,

where

J An (s) :=

∫0<sn<....<s1<s

Λsn dWAsn Λsn−1−sn . . . dW

As2 Λs1−s2 dW

As1 Λs−s1

=∑

1≤j1,...,jn≤k

∫0<sn<...<s1<s

Λsn Ajn Λsn−1−sn . . . Aj2 Λs1−s2 Aj1 Λs−s1 dWj1s1 . . . dW

jnsn ,

and where the remainder J An (s)X is obtained by replacing the term Λsn by Xsn , in theabove integral defining J An (s).

Indeed, for N = 1 this is the above formula (∗), and if this holds for some N ≥ 1, then, usingFormula (∗) again, we obtain :

J AN (s)X =

∫0<sN<...<s1<s

[ΛsN+

∫0<sN+1<sN

XsN+1dWAsN+1

ΛsN−sN+1

]dWA

sNΛsN−1−sN . . .Λs−s1

= J AN (s) + J AN+1(s)X .

Then using Lemma X.2.1, Lemma VII.2.12, and Remark VII.2.13 repeatedly, we have :

E[∥∥∥J An (s)

∥∥∥2

HS

]

=∑

1≤j1,..., jn≤k

∫0<sn<...<s1<s

∥∥∥Λsn Ajn Λsn−1−sn . . . Aj2 Λs1−s2 Aj1 Λs−s1

∥∥∥2

HSdsn . . . ds1

≤∑

1≤j1,...,jn≤k

∥∥Aj1∥∥2

HS. . .∥∥Ajn∥∥2

HS

∫0<sn<...<s1<s

∥∥Λsn∥∥2

HS. . .∥∥Λs−s1

∥∥2

HSdsn . . . ds1 .

Observe that using Lemma VII.2.12, for any s ≥ 0 we have :

∥∥Λs∥∥HS≤∑n≥0

sn

n!

∥∥Bn∥∥HS≤ es ‖B‖HS ≤ exp

[(∥∥A0

∥∥HS

+ 12

k∑j=1

∥∥Aj∥∥2

HS

)s

],

X.3. BROWNIAN PATH AND LIMITING GEODESIC 321

whence

E[∥∥∥J An (s)

∥∥∥2

HS

]≤

∑1≤j1,...,jn≤k

‖Aj1‖2HS . . . ‖Ajn‖2HSsn

n!e2s‖B‖HS =

sn

n!

k∑j=1

∥∥Aj∥∥2

HS

ne2s‖B‖HS .

This proves the convergence of the series in L2-norm, uniformly for bounded s .

Finally, we deduce from the preceding and from Lemma VII.2.12 the following control of theremainder in the induction formula :

E[∥∥∥J AN (s)X

∥∥∥2

HS

]≤

k∑j=1

∥∥Aj∥∥2

HS

N ∫0<sN<...<s1<s

e2 ‖B‖HS (s−sN ) E[∥∥XsN

∥∥2

HS

]dsN . . . ds1

≤ d

N !

k∑j=1

∥∥Aj∥∥2

HS× s

N × exp

[2(∥∥A0

∥∥HS

+k∑j=1

∥∥Aj∥∥2

HS

)s

]−−−→N→∞

0 .

X.3 Brownian path and limiting geodesic

We saw in Proposition VII.6.7.3 that the hyperbolic Brownian motion (zs)(started at

p ∈ Hd) converges almost surely (as s → ∞) to a boundary point z∞ ∈ ∂Hd, the law ofwhich is the harmonic measure µp . Here we specify the asymptotic proximity between thetypical Brownian path and its almost sure limiting geodesic (through p , ending at z∞).

Let us begin with the disintegration of the law of the hyperbolic Brownian motion withrespect to its end-point z∞ ∈ ∂Hd.

Proposition X.3.1 The law of the hyperbolic Brownian motion (Zs) started at Z0 = p ∈ Hd

disintegrates with respect to its end-point z∞ ∈ ∂Hd as follows : for any positive s andbounded measurable functions ϕ on ∂Hd and F on the space of trajectories (started from p)Cp([0, s],Hd), we have

Ep[F (z[0,s])ϕ(z∞)

]=

∫∂Hd

Ee0[F g−1

η

(z[0,s]

)]ϕ(η)µe0(dη) ,

where gη ∈ PSO(1, d) denotes any isometry such that gη p = e0 and gη η = e0 + e1 , and(zs)s≥0 , whose law is described in Poincare coordinates

(xs, ys

) (using a standard Euclidean

Brownian motion (W 1t ,W

′t) ∈ R×Rd−1) by :

ys = eW1s +(d−1)s/2 and xs =

∫ s

0yt dW

′t ,


denotes the hyperbolic Brownian motion conditioned to exit Hd at R+(e0 +e1) ∈ ∂Hd, and isalso the process already encountered as associated with the dual process

(Z∗s = Tzs

)of Section

VIII.1.

Proof Using that the law of z∞ conditionally on zs is the harmonic measure µzs , FubiniTheorem and Proposition III.6.1.5, we have the following generalization of

([F3], Section 14.1,

Lemma 8)

:

Ep[F (z[0,s])ϕ(z∞)

]=

∫∂Hd

Ep[F (z[0,s])ϕ(η)µzs(dη)

]=

∫∂Hd

Ep[F (z[0,s]) 〈zs, ηp〉1−d

]ϕ(η)µp(dη) =

∫∂Hd

Ep[F (z[0,s]) e

(d−1)Bη(p,zs)]ϕ(η)µp(dη) .

This means that the law of the hyperbolic Brownian motion started from p , conditionally onthe event that its end-point is z∞ ∈ ∂Hd, admits the density 〈zs, (z∞)p〉1−d = e(d−1)Bz∞ (p,zs)

with respect to its unconditioned law. 1

By the invariance of the Brownian law under the isometry gη ∈ PSO(1, d) (chosen as in thestatement), we then obtain

Ep[F (z[0,s]) e

(d−1)Bη(p,zs)]

= Ee0[F(g−1η z[0,s]

)e(d−1)Bη(p,g−1

η zs)]

= Ee0[F g−1

η

(z[0,s]

)e(d−1)Bgη η(gη p , zs)

]= Ee0

[F g−1

η

(z[0,s]

)e(d−1)Be0+e1 (e0,zs)

]= Ee0

[F g−1

η

(z[0,s]

) ⟨zs, e0 + e1

⟩1−d].

Otherwise, denoting by (xs, ys) the Poincare coordinates of (zs) and using Formula (I.18), wehave :

zs =(y2

s + |xs|2 + 1

2 ys

)e0 +

(y2s + |xs|2 − 1

2 ys

)e1 +

d∑j=2

xjsysej ,

whence⟨zs , e0 + e1

⟩= y−1

s . So far we obtain :

Ep[F (z[0,s])ϕ(z∞)

]=

∫∂Hd

Ee0[F g−1

η

(z[0,s]

)yd−1s

]ϕ(η)µe0(dη)

=

∫∂Hd

Ee0[F g−1

η

(z[0,s]

)]ϕ(η)µe0(dη) ,

where (zs) denotes the hyperbolic Brownian motion conditioned to exit Hd at R+(e0 + e1).

1In other words, the conditioned hyperbolic Brownian motion is a so-called “h-process” (in thesense of [Do]) of the unconditioned one, the harmonic function h being here the Poisson kernelappearing in Proposition III.6.1.4.

X.3. BROWNIAN PATH AND LIMITING GEODESIC 323

Denoting by(xs, ys

)its Poincare coordinates, from the above we deduce that :

E(x,y)

[f(xs, ys)

]= E(x,y)

[f(xs, ys)(ys/y)d−1

]= y1−d Ps

(yd−1 f

)(x, y) = P ∗s f(x, y) ,

according to Remark VIII.1.1(ii). Hence the law of (zs) is computed in Poincare coordinates(xs, ys

)by the expression (VIII.4) of the Poincare coordinates associated to the dual process(

Z∗s = Tzs)

of Section VIII.1, which are precisely the expressions of the statement. Thisproves the proposition.

Of course the above proposition X.3.1 is valid with any test-functional F(z[0,s], z∞

)as

well. We specialize now to the functional dist (zs, γ∞), where γ∞ denotes the randomlimiting geodesic determined by p and z∞ . See Figure X.1. We can also consider the verysimply related horocyclic distance functional distH(zs, γ∞), as specified by the following (thehorocyclic distance between two points was already encountered page 65).

Lemma X.3.2 Given an oriented geodesic γ∞ ≡ (η′, η) and z ∈ Hd, the horocyclic distancedistH(z, γ∞), i.e., the hyperbolic length of the minimal curve which links z to γ∞ ∩ Hη(z)within the horocycle Hη(z), is equal to the hyperbolic sinus sh

[dist (z, γ∞)

]of the hyperbolic

distance (and thus does not depend on the orientation of the geodesic γ∞).

Proof Consider (as in Proposition X.3.1) an isometry gη ∈ PSO(1, d) such that gη η′ =

e0 − e1 and gη η = e0 + e1 , so that γ′∞ := gη γ∞ is the vertical geodesic (0,∞) in Poincarecoordinates. Denote by (x, y) the Poincare coordinates of z′ := gη z . Then on the one handwe have

distH(z, γ∞) = distH(z′, γ′∞) = |x|/y .On the other hand, using that the minimizing geodesic from z′ to γ′∞ is orthogonal to γ′∞(recall Proposition II.2.2.4) and Formula (II.1), we find at once that :

dist (z, γ∞) = dist (z′, γ′∞) = dist(

(x, y) ;(0,√|x|2 + y2

))= argch

[√|x|2 + y2

y

]= argsh

[|x|/y

]= argsh

[distH(z, γ∞)

].

The asymptotic behaviour of dist (zs, γ∞) is given by the following.

Proposition X.3.3 Let γ∞ denote the random limiting geodesic determined by p andz∞ . Then as s → ∞ , the horocyclic distance distH(zs, γ∞)

(i.e., the hyperbolic sinus

sh[dist (zs, γ∞)

]of the hyperbolic distance

)from the hyperbolic Brownian motion zs to γ∞

converges in law to the Euclidean norm under the harmonic measure µe0 . Hence it admitsthe limiting density (with respect to the Lebesgue measure of R+) :

r 7−→ 2d−1 Γ(d/2)√π Γ( d−1

2)

(1 + r2)1−d rd−2 .


Figure X.1: Brownian path and limiting geodesic

Proof We use the above proposition X.3.1. Denote by γ′∞ the vertical geodesic throughe0 (in the canonical frame). By the invariance of the hyperbolic metric under the isometrygη , we obtain

distH(g−1η zs, γ∞) = distH(zs, gηγ∞) = distH(zs, γ

′∞) = |xs|

/ys .

Hence applying Proposition X.3.1 we obtain : for any bounded continuous function f ,

Ep[f(distH(zs, γ∞)

)]=

∫∂Hd

Ee0[f(distH(zs, γ

′∞))]µe0(dη) = E

[f(|xs|/ys)].

Thus it is sufficient to specify the asymptotic law of xs/ys . Now changing the variable t

into (s− t) we obtain the following easy identity in law :

xsys

=

∫ s

0eW

1t −W 1

s + d−12

(t−s) dW ′t ≡∫ s

0eW

1t −

d−12t dW ′t ,

and the latter converges almost surely to x∞ :=

∫ ∞0

eW1t −

d−12t dW ′t , according to the proof

of Proposition VII.6.7.3. Moreover by this same proof we almost surely also have xs −→ x∞ ,so that x∞ is equal to the non-trivial Poincare coordinate of the end-point z∞ , when thehyperbolic Brownian motion (zs) is started from e0 . Therefore the law of x∞ is given bythe harmonic measure µe0 , whose density with respect to the Lebesgue measure of Rd−1

is[x 7−→ 2d−2 Γ(d/2)

πd/2(1 + |x|2)1−d

], according to Lemma III.6.1.2. The limiting density of

distH(zs, γ′∞) is easily deduced, as that of |x∞| .

Chapter XI

Index of notation, terms, andfigures

XI.1 General notation

XI.2 Other notation

XI.3 Index of terms

XI.4 Table of figures

325

326 CHAPTER XI. INDEX OF NOTATION, TERMS, AND FIGURES

References

[A] Ancona A. Theorie du potentiel sur les graphes et les varietes.

Ecole d’ete de prob. de Saint-Flour XVIII, Lecture Notes 1427, Springer 1990.

[B1] Bowditch B.H. Geometrical finiteness for hyperbolic groups.

J. Funct. Anal. 113, 245-317, 1993.

[B2] Bowditch B.H. Geometrical finiteness with variable negative curvature.

Duke Math. J. (1) 77, 229-274, 1995.

[Ba] Baxendale P.H. Asymptotic behaviour of stochastic flows of diffeomorphisms :

two case studies. Prob. Th. Rel. Fields 73, 51-85, 1986.

[Be] Bernstein S. Equation differentielles stochastiques.

Act. Sci. Ind. 738, Conf. Math. Univ. Geneve, 5-31, Hermann, Paris 1938.

[Bi] Billingsley P. Probability and measure. Wiley Series Prob. Math. Stat.

Wiley-Intersc. Publ. John Wiley & Sons, New York 1979, 1986, 1995.

[BF] Bailleul I., Franchi J. Non-explosion criteria for relativistic diffusions.

ArXiv : http://arxiv.org/abs/1007.1893. To appear at Annals of Probability.

[Bk] Bourbaki N. Elements de mathematique. Fasc. XXXVII. Groupes et algebres

de Lie. Act. Scient. Indust. nos 1285 and 1349, Hermann, Paris 1971-72.

[Bl] Bailleul I. Poisson boundary of a relativistic diffusion.

Prob. Th. Rel. Fields vol. 141, no 1-2, 283-329, 2008.

[Bn] Bourdon M. Structure conforme au bord et flot geodesique d’un CAT(-1)-espace.

L’enseignement Mathematique, tome 41, 63-102, 1995.

[By] Bakry D. Functional inequalities for Markov semigroups.

Probability measures on groups: recent directions and trends, 91-147,

Tata Inst. Fund. Res., Mumbai 2006.

REFERENCES 327

[BP1] Babillot M, Peigne M. Homologie des geodesiques fermees sur des varietes

hyperboliques avec bouts cuspidaux. Ann. Sc. E.N.S. (4) 33, 81-120, 2000.

[BP2] Babillot M., Peigne M. Asymptotic laws for geodesic homology on hyperbolic

manifolds with cusps. Bull. Soc. Math. France 134, no 1, 119-163, 2006.

[CE1] Carverhill A.P., Elworthy K.D. Flows of stochastic dynamical systems : the

functional analytic approach. Z. Wahrsch. Verw. Geb. 65, no 2, 245-267, 1983.

[CE2] Carverhill A.P., Elworthy K.D. Lyapunov exponents for a stochastic analogue

of the geodesic flow. Trans. Amer. Math. Soc. 295, no 1, 85-105, 1986.

[CFS] Cornfeld I.P., Fomin S.V., Sinai Y.G. Ergodic theory.

Grundl. Math. Wissensch. 245, Springer, New York 1982.

[CG] Chen S.S., Greenberg L. Hyperbolic spaces.

Contributions to analysis (a collection of papers dedicated to Lipman Bers),

49-87. Academic Press, New York 1974.

[CK] Carmona R., Klein A. Exponential Moments for Hitting Times of Uniformly

Ergodic Markov Processes. Annals of Probability 11, no 3, 648-655, 1983.

[CL] Conze J.P., Leborgne S. Methode de martingales et flot geodesique sur une

surface de courbure constante negative. Erg. Th. Dyn. Syst. 21, no 2, 421-441, 2001.

[Da] Dal’bo F. Trajectoires geodesiques et horocycliques.

Savoirs actuels, EDP Sciences, CNRS Editions, Paris 2007.

[Deb] Debbasch F. A diffusion process in curved space-time.

J. Math. Phys. 45, no 7, 2744-2760, 2004.

[Do] Doob J.L. Classical potential theory and its probabilistic counterpart.

Springer, New York 1984, Berlin 2001.

[D1] Dudley R.M. Lorentz-invariant Markov processes in relativistic phase space.

Arkiv for Matematik 6, no 14, 241-268, 1965.

[D2] Dudley R.M. A note on Lorentz-invariant Markov processes.

Arkiv for Matematik 6, no 30, 575-581, 1967.


[D3] Dudley R.M. Asymptotics of some relativistic Markov processes.

Proc. Nat. Acad. Sci. USA no 70, 3551-3555, 1973.

[Dt] Durrett R. Probability : theory and examples. Wadsworth-Brooks/Cole, Pacific

Grove CA 1991 ; Duxbury Pr., Belmont CA 1996 ; Camb. Univ.Pr., Cambridge 2010.

[DFN] Dubrovin B.A., Fomenko A.T., Novikov S.P. Modern geometry ; methods

and applications. Graduate Texts in Math. 93. Springer, New York 1984, 1992.

[E] Emery M. On some relativistic-covariant stochastic processes in Lorentzian

space-times. C. R. Math. Acad. Sci. Paris 347, no 13-14, 817-820, 2009.

[EF] Enriquez N., Franchi J. Masse des pointes, temps de retour et enroulements

en courbure negative. Bull. Soc. Math. France 130, no 3, 349-386, 2002.

[EFLJ1] Enriquez N., Franchi J., Le Jan Y. Stable windings on hyperbolic surfaces.

Prob. Th. Rel. Fields, vol. 119, 213-255, 2001.

[EFLJ2] Enriquez N., Franchi J., Le Jan Y. Canonical lift and exit law of the

fundamental diffusion associated with a Kleinian group.

Sem. Prob. XXXV, 206-219, Springer 2001.

[EFLJ3] Enriquez N., Franchi J., Le Jan Y. Central limit theorem for the geodesic

flow associated with a Kleinian group, case δ > d/2.

J. Math. Pures Appl. 80, no 2, 153-175, 2001.

[EFLJ4] Enriquez N., Franchi J., Le Jan Y. Enroulements browniens et subordination

dans les groupes de Lie. Seminaire de Probabilites XXXIX, 357-380, Springer 2005.

[ELJ] Enriquez N., Le Jan Y. Statistic of the winding of geodesics on a Riemann

surface with finite volume and constant negative curvature.

Rev. Mat. Iberoam. (2) 13, 377-401, 1997.

[F1] Franchi J. Asymptotic singular windings of ergodic diffusions.

Stochastic Processes and their Applications, vol. 62, 277-298, 1996.

[F2] Franchi J. Asymptotic singular homology of a complete hyperbolic 3-manifold of

finite volume. Proc. London Math. Soc. (3) 79, 451-480, 1999.

REFERENCES 329

[F3] Franchi J. Asymptotic windings over the trefoil knot.

Rev. Mat. Iberoam., vol. 21, no 3, 729-770, 2005.

[F4] Franchi J. Relativistic Diffusion in Godel’s Universe.

Comm. Math. Physics, vol. 290, no 2, 523-555, 2009.

[FK] Fricke R., Klein F. Vorlesungen uber die Theorie der automorphen Funktionen.

Band I : Die gruppentheoretischen Grundlagen.

Band II : Die funktionentheoretischen Ausfhrungen und die Andwendungen.

Bibl. Math. Teubneriana, Bande 3, 4 Johnson Reprint Corp., New York ;

B. G. Teubner Verlagsgesellschaft, Stuttgart 1965.

[FgK] Furstenberg H., Kesten H. Products of random matrices.

Ann. Math. Statist. vol. 31, 457-469, 1960.

[FLJ1] Franchi J., Le Jan Y. Relativistic Diffusions and Schwarzschild Geometry.

Comm. Pure Appl. Math. vol. LX, no 2, 187-251, 2007.

[FLJ2] Franchi J., Le Jan Y. Curvature Diffusions in General Relativity.

Comm. Math. Physics vol. 307, no 2, 351-382, 2011.

[Gh] Ghys E. Poincare and his disk. The scientific legacy of Poincare, 17-46,

Hist. Math. 36, Amer. Math. Soc., Providence, RI 2010.

[Gi1] Gihman I.I. On some differential equations with random functions.

(Russian.) Ukrain. Mat. Zurnal 2, no 3, 45-69, 1950.

[Gi2] Gihman I.I. On the theory of differential equations of random processes.

(Russian.) Ukrain. Mat. Zurnal 2, no 4, 37-63, 1950.

[GLJ] Guivarc’h Y., Le Jan Y. Asymptotic windings of the geodesic flow on modular

surfaces with continuous fractions. Ann. Sci. E.N.S. 26, no 4, 23-50, 1993.

[Gr] Gromov M. Structures Metriques pour les Varietes Riemanniennes.

Redige par J. Lafontaine et P. Pansu. Cedic-F. Nathan, Paris 1981.

[GS] Gihman I.I., Skorohod A.V. The theory of stochastic processes. I, II, III.

Springer, Berlin 2004, 2007.


[Gu] Guivarc’h Y. Proprietes de melange pour les groupes a un parametre de Sl(d,R).

Fasc. probab., 11 pp., Publ. Inst. Rech. Math. Rennes, Univ. Rennes I, 1992.

[Ha] Has’minskiı R.Z. Stochastic stability of differential equations.

(Transl. from Russian) Monog. Textb. Mech. Sol. Fluids : Mech. Anal. 7,

Sijthoff & Noordhoff, Alphen aan den Rijn, Germantown, Md., 1980.

[HE] Hawking S.W., Ellis G.F.R. The large-scale structure of space-time.

Cambrige Univ. Press, Cambrige1973.

[He] Helgason S. Differential geometry and symmetric spaces.

Pure Appl. Math.Vol. XII, Ac. Press, first edition, New York-London 1962.

[HK] Hasselblatt B., Katok A. Introduction to the modern theory of dynamical

systems. Encycl. Math. Appl. 54, Cambridge Univ. Pr., Cambridge 1995.

[Ho] Hochschild G. The structure of Lie groups. Holden-Day, Inc., 1965.

[Hop] Hopf E. Ergodicity theory and the geodesic flow on a surface of constant

negative curvature. Bull. Amer. Math. Soc. 77, 863-877, 1971.

[I] Ito K. Differential equations determining Markov processes. (Japanese.)

Zenkoku Shijo Sugaku Danwakai 244, no 1077, 1352-1400, 1942.

[IMK] Ito K., McKean H.P. Diffusion processes and their sample paths.

Sec. pr., correct. Grundl. math. Wiss. 125, Springer, Berlin-New York 1974.

[IW] Ikeda N., Watanabe S. Stochastic differential equations and diffusion processes.

North-Holland Kodansha, 1981.

[J] Jacod J. Calcul stochastique et problemes de martingales.

Lecture Notes in Math. vol. 714, Springer, Berlin 1979.

[Ka] Kallenberg O. Foundations of Modern Probability. Springer, Berlin 1997, 2002.

[Kn] Knapp A.W. Lie Groups Beyond an Introduction.

Progress in Mathematics vol. 140, Birkhauser, Boston Basel Berlin 2002.

REFERENCES 331

[KN] Kobayashi S., Nomizu K. Foundations of Differential Geometry.

Interscience Publishers, John Wiley & Sons, New York-London-Sydney, 1969.

[Kr] Kra I. Automorphic forms and Kleinian groups.

Math. Lect. Note Series. W. A. Benjamin, Inc., Reading, Mass., 1972.

[KS] Karatzas I., Shreve S. Brownian motion and stochastic calculus.

Graduate Texts in Math. 113. Springer-Verlag, New York 1988, 1991.

[Ku] Kunita H. Stochastic flows and stochastic differential equations.

Cambridge Stud. in Adv. Math. 24, Cambr. Univ. Pr., Cambridge 1990, 1997.

[L1] Ledrappier F. Quelques proprietes des exposants caracteristiques.

Ecole d’ete prob. St-Flour XII-1982, 305-396, L. N. 1097, Springer, Berlin 1984.

[L2] Ledrappier F. Harmonic 1-forms on the stable foliation.

Bol. Soc. Bras. Math. 25, no 2, 121-138, 1994.

[L3] Ledrappier F. Central limit theorem in negative curvature.

Annals of Probability 23, no 3, 1219-1233, 1995.

[Le] Lehner J. Discontinuous groups and automorphic functions.

Amer. Math. Soc., Math. Survey no VIII, Providence, 1964.

[Lev] Levy P. Processus Stochastiques et Mouvement Brownien. Suivi d’une note

de M. Loeve. Gauthier-Villars, Paris 1948.

[LJ1] Le Jan Y. Sur l’enroulement geodesique des surfaces de Riemann.

C.R.A.S. Paris, vol 314, Serie I, 763-765, 1992.

[LJ2] Le Jan Y. The central limit theorem for the geodesic flow on non compact

manifolds of constant negative curvature. Duke Math. J. (1) 74, 159-175, 1994.

[LJ3] Le Jan Y. Free energy for Brownian and geodesic homology.

Probab. Theory Rel. Fields 102, 57-61, 1995.

[LJ4] Le Jan Y. On isotropic Brownian motions.

ex Z.f.W. then Probab. Theory Rel. Fields 70, 609-620, 1985.


[LJW] Le Jan Y., Watanabe S. Stochastic flows of diffeomorphisms. Stoch. anal.,

Katata/Kyoto 1982, 307-332, North-Holland Math. Lib. 32, Amsterdam 1984.

[LL] Landau L., Lifchitz E. Physique theorique, tome II : Theorie des champs.

Editions MIR de Moscou, 1970.

[Mar] Margulis G.A. Discrete subgroups of semisimple Lie groups.

Ergeb. Math. Grenzgeb. (3) 17, Springer, Berlin 1991.

[Mas] Maskit B. Kleinian groups.

Grundl. Math. Wissensch. 287, Springer, Berlin 1988.

[Me] Meyer P.-A. Un cours sur les integrales stochastiques.

Sem. Probab. X (Sec. partie : Theorie des integrales stochastiques, Univ.

Strasbourg, 1974-75), 245-400. Lect. Notes Math. vol. 511, Springer, Berlin 1976.

[Mi] Miyake T. Modular forms. Springer, Berlin 1989.

[MK] McKean H.P. Stochastic Integrals. Academic Press, New York & London, 1969.

[Mr] Milnor J. Hyperbolic geometry : the first 150 years.

Bull. Amer. Math. Soc. (N.S.) 6, no 1, 9-24, 1982.

[MT] Matsuzaki K., Taniguchi M. Hyperbolic manifolds and Kleinian groups.

Oxf. Math. Mon., Clarendon Press, Oxford Univ. Pr., New York 1998.

[N1] Neveu J. Martingales a temps discret. Masson et Cie, Paris 1972.

[N2] Neveu J. Courte demonstration du theoreme ergodique sur-additif.

Ann. Inst. H. Poincare, vol. XIX, no 1, 87-90, 1983.

[Ni] Nicholls P.J. The ergodic theory of discrete groups.

London Math. Soc., Lecture Note Series no 143, Cambridge University Press, 1989.

[0] Oksendal B. Stochastic differential equations. An introduction with applications.

Universitext. Springer, Berlin 1985, 1989, 1992, 1995, 1998, 2003.

[0s] Oseledets V.I. A multiplicative ergodic theorem. Characteristic Ljapunov expo-

nents of dynamical systems. (Russian) Trudy Moskov. Mat. Obsc. 19, 179-210, 1968.

REFERENCES 333

[Pa] Pansu P. Metriques de Carnot-Caratheodory et quasiisometries des espaces

symetriques de rang un. Annals of Math. (2) 129, no 1, 1-60, 1989.

[P1] Patterson S.J. The Laplacian operator on a Riemann surface, III.

Compositio Mathematica vol. 33, Fasc. 3, 227-259, 1976.

[P2] Patterson S.J. Lectures on measures on limit sets of Kleinian groups.

Analytical and geometrical aspects of hyperbolic space, D. Epstein editor, 281-323,

London Math. Society, Lecture Note S. no 111, Cambridge University Press, 1987.

[Pr] Protter J. Stochastic integration and differential equations.

Stoch. Model. Appl. Prob. 21, Springer, Berlin 1990, 2005.

[PS] Phillips R.S., Sarnak P. The Laplacean for domains in hyperbolic space and

limit sets of Kleinian groups. Acta Math. 155, 173-241, 1985.

[PSt] Port S.C., Stone C.J. Brownian motion and classical potential theory.

Prob. Math. Stat., Acad. Press, New York-London 1978.

[Rac] Ratcliffe J.G. Foundations of hyperbolic manifolds. Springer, Berlin 1994.

[Rag] Raghunathan M.S. Discrete subgroups of Lie groups.

Ergeb. Math. Grenzgeb. 68, Springer, New York-Heidelberg 1972.

[Ran] Ratner M. The central limit theorem for geodesic flows on n-dimensional

manifolds of negative curvature. Israel J. of Math. 16, 181-197, 1973.

[Ru] Ruelle D. Ergodic theory of differentiable dynamical systems.

Inst. Hautes Etudes Sci. Publ. Math. no 50, 27-58, 1979.

[RY] Revuz D., Yor M. Continuous martingales and Brownian motion.

Springer, Berlin 1991, 1994, 1999.

[RW1] Rogers L.C.G., Williams D. Diffusions, Markov processes, and martingales.

Vol. 1. Foundations. John Wiley & Sons, New York-London-Sydney, 1994.

[RW2] Rogers L.C.G., Williams D. Diffusions, Markov processes, and martingales.

Vol. 2. Ito calculus. Cambr. Math. Lib., Cambr. Univ. Pr., Cambridge 2000.


[Sc] Schoeneberg B. Elliptic modular functions. Springer, Berlin 1974.

[SG] de Saint-Gervais H.P. Uniformisation des surfaces de Riemann. Retour sur

un theoreme centenaire. E.N.S. Editions, Lyon 2010.

[Si1] Sinai Y.G. The central limit theorem for geodesic flows on manifolds of constant

negative curvature. Dokl. Akad. Nauk. SSSR 133, 1303-1306, 1960.

[Si2] Sinai Y.G. The central limit theorem for geodesic flows on manifolds of constant

negative curvature. Soviet Math. Dokl. 1, 938-987, 1960.

[St] Steele J.M. Kingman’s subadditive ergodic theorem.

Ann. I.H.P., section B, tome 25, no 1, 93-98, 1989.

[Su] Sullivan D. Entropy, Hausdorff measures old and new, and limit sets of

geometrically finite Kleinian groups. Acta Math. 153, 259-277, 1984.

[Sw] Swan R. G. Generators and relations for certain special linear groups.

Adv. Math. 6, 1-77, 1971.

[W] Wald R.M. General relativity. University of Chicago Press, Chicago IL 1984.

[Y] Yue C. The ergodic theory of discrete isometry groups on manifolds of variable

negative curvature. Trans. A. M. S. 348, no 12, 4965-5005, 1996.

——————————————————————————————–

Jacques FRANCHI : Universite de Strasbourg et CNRS, I.R.M.A., 7 rue Rene

Descartes, 67084 Strasbourg cedex. FRANCE.

[email protected]

Yves LE JAN : Universite Paris Sud, Mathematiques, Batiment 425, 91405 Orsay.

FRANCE.

[email protected]

——————————————————————————————–

hyperbolic dynamics - unistra.frirma.math.unistra.fr/~franchi/livre'.pdf · 2012-02-13 ·...

Documents