Contents

Introduction 3

1 Preliminaries 71.1 Topological groups . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . 91.3 Cartan decompositions . . . . . . . . . . . . . . . . . . . . . . 131.4 Some basics of Riemannian geometry . . . . . . . . . . . . . . 171.5 Some functional analysis . . . . . . . . . . . . . . . . . . . . . 20

1.5.1 Fréchet spaces . . . . . . . . . . . . . . . . . . . . . . 201.5.2 Fréchet G-modules . . . . . . . . . . . . . . . . . . . . 231.5.3 Integration of vector-valued functions . . . . . . . . . 25

2 Symmetric Spaces 272.1 Globally symmetric spaces . . . . . . . . . . . . . . . . . . . . 272.2 Decomposition of symmetric spaces . . . . . . . . . . . . . . . 332.3 Curvature of symmetric spaces . . . . . . . . . . . . . . . . . 352.4 Maximal compact subgroups . . . . . . . . . . . . . . . . . . . 37

3 Continuous Group Cohomology 433.1 Continuous Group Cohomology . . . . . . . . . . . . . . . . . 433.2 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3 Discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 Van Est’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Bounded Cohomology 614.1 Continuous bounded cohomology . . . . . . . . . . . . . . . . 614.2 Simplicial volume and bounded cohomology . . . . . . . . . . 704.3 Smooth homology and cohomology . . . . . . . . . . . . . . . 764.4 Other resolutions . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.1 Continuous maps on coset spaces . . . . . . . . . . . . 794.4.2 Measurable maps on coset spaces . . . . . . . . . . . . 824.4.3 Discrete subgroups . . . . . . . . . . . . . . . . . . . . 884.4.4 Alternating maps . . . . . . . . . . . . . . . . . . . . . 90

4.5 Lattices in Lie groups . . . . . . . . . . . . . . . . . . . . . . 94

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4.6 Cup product . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Gromov’s Proportionality Principle 1055.1 A proof of Gromov’s Proportionality Principle . . . . . . . . . 1055.2 The proportionality constant for locally symmetric spaces . . 113

6 Manifolds Covered By H2 ×H2 1176.1 Sectional curvature of H2 ×H2 . . . . . . . . . . . . . . . . . 1176.2 The isometry group of the hyperbolic plane . . . . . . . . . . 1206.3 The isometry group of H2 ×H2 . . . . . . . . . . . . . . . . . 1226.4 Proof of Theorem 6.3.6 . . . . . . . . . . . . . . . . . . . . . . 1276.5 Volume of straight simplices in H2 ×H2 . . . . . . . . . . . . 131

Bibliography 135

2

Introduction

The main purpose of this dissertation is to introduce the concept of simplicialvolume of a closed manifold and to study it for locally symmetric spacesfollowing [BK08a]; we will also present explicit calculations in the case ofmanifolds covered by Hn or H2×H2, following [Thu78], [Gro82] and [BK08b].The notion of simplicial volume of a manifold Mn was first introduced byGromov in his seminal paper “Volume and bounded cohomology” in 1982; ithas been widely studied since, due to the number of applications in a varietyof branches of geometry.Being defined as the `1-norm of the fundamental class of M in its n-th realhomology space, the simplicial volume ‖M‖ is a purely topological invariant;when ‖M‖ > 0, it provides an obstruction to the existence of maps from agiven manifold to M with arbitrarily large degree (see Lemma 4.2.3).Nevertheless, the simplicial volume can often capture geometric propertiesof Riemannian manifolds. For example, Gromov originally introduced thisconcept to study the minimal volume of manifolds, i.e.:

MinVol M := infvol(M, g) | | sec(M, g)| ≤ 1 ;

namely he showed that, for arbitrary manifolds, the following inequalityholds (see Corollary A in §0.5 in [Gro82]):

‖M‖ ≤ (n− 1)nn! ·MinVol M .

Moreover, Gromov’s Proportionality Principle surprisingly binds the simpli-cial volume to the usual Riemannian volume (a purely geometrical notion)for closed manifolds with isometric universal coverings (see Chapter 5).Furthermore, the possible Euler numbers of n-dimensional flat vector bun-dles (i.e. admitting a set of trivialisations with constant transition functions)over M are related to ‖M‖. This was originally observed in 1958 by Milnorin the case of bundles over surfaces (see [Mil58]) and later extended to amore general setting by Sullivan ([Sul76], [Gro82]) and Smillie ([Smi]) whoproved:

‖M‖ ≥ 2n|χ(E)|

for all n-dimensional flat vector bundles E →M (for details, see also [IT85]and [BKM12]).

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These results provide the most interesting consequences in the case of non-vanishing simplicial volume. This happens for a wide variety of manifolds,such as negatively curved closed manifolds, closed locally symmetric spacesof noncompact type, products of these examples and connected sums of thesewith arbitrary manifolds, in dimension at least 3 (see Chapter 5). However,‖M‖ vanishes whenever π1(M) is amenable or, in particular, solvable.In Chapters 1 and 2 we will recall some standard notions that will be use-ful in the subsequent treatment. In particular, Chapter 2 will be devotedto the general theory of symmetric spaces, which is closely related to thestudy of semisimple Lie groups and Lie algebras. The fact that simply con-nected symmetric spaces decompose in factors with nonpositive, nonnegativeor vanishing curvature will be the main motivation for our restricted interestto the simplicial volume of these spaces.In Chapter 3 we will introduce the continuous cohomology spaces of a topo-logical group G; this can be seen as a generalisation of the usual groupcohomology, in that both theories yield the same result for countable dis-crete groups. This concept was first defined by S. Hu in 1954 ([Hu54]) andfurther studied by Van Est and Mostow who explored its relationship withthe theory of Lie algebra cohomology of Chevalley and Eilenberg (see [Est55],[Mos61]). The categorical approach via relatively injective modules that wepresent is due to Hochschild and Mostow, who systematised the theory in1962 ([HM62]); we will follow the treatment in Guichardet’s book ([Gui80]).When G is the isometry group of a contractible symmetric space S, thecontinuous cohomology of G will prove a useful tool, being related to thespace of G-invariant differential forms on S by the Van Est isomorphism.We will also widely exploit the connections between continuous cohomologyand continuous bounded cohomology, which we will define in Chapter 4.Bounded cohomology was first introduced for topological spaces and discretegroups by Gromov in 1982 ([Gro82]), as a tool for studying the simplicialvolume of manifolds. This theory was then extended to topological groups byBurger and Monod, who defined continuous bounded cohomology in [Mon01].The resulting interplay between the simplicial volume of closed locally sym-metric spaces, the cohomology and bounded cohomology of their fundamen-tal groups, the continuous cohomology and continuous bounded cohomologyof the isometry group of their universal coverings will be decribed in Chap-ter 4 and it will be at the core of the results in Chapters 5 and 6.In particular, after the proof of Gromov’s Proportionality Principle followingThurston ([Thu78]) and Löh ([L06]), Chapter 5 will be devoted to under-standing for which locally symmetric spaces the simplicial volume vanishes.We will also give an explicit computation of the simplicial volume of hy-perbolic manifolds, result originally due to Gromov and Thurston (see forexample [BP92] or [Thu78]).Finally, Chapter 6 will be entirely devoted to Bucher-Karlsson’s computa-tion of the simplicial volume of manifolds covered by H2 × H2. We remark

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that, to this day, this remains the only class of manifolds (except hyperbolicones and those that can be obtained with Theorem 5.2.2) for which the exactvalue of the simplicial volume is known and nonzero.Moreover, we will see that the ratio between the Riemannian volume and thesimplicial volume of closed, oriented manifolds covered by H2 × H2 differsfrom the supremum of volumes of straight simplices in the universal coveringspace. This is a previously unnoticed and somewhat astonishing differencefrom the hyperbolic case.

5

6

Chapter 1

Preliminaries

1.1 Topological groups

Definition 1.1.1. A topological group is a group G which is also equippedwith a topology such that the map

G×G→ G

(x, y) 7→ x−1y

is continuous.

In particular we have homeomorphisms:

Lg : G→ G Rg : G→ G Cg : G→ G

x 7→ gx x 7→ xg x 7→ gxg−1

which we will keep denoting this way throughout the dissertation. Moreover,we also define the homeomorphisms

Lkg : Gk → Gk Rkg : Gk → Gk

(g0, ..., gk) 7→ (gg0, ..., ggk) (g0, ..., gk) 7→ (g0g, ..., gkg)

for later use. We will refer to Lg and Rg as left translations and righttranslations, respectively.A discrete, normal subgroup Γ of a connected topological group G is alwayscontained in the centre of G. The next lemma easily follows from this fact.

Lemma 1.1.2. Let p : G → H be a covering map between connected topo-logical groups, which is also a homomorphism. Then the kernel of p is adiscrete subgroup of the centre of G.

Remark 1.1.3. The notion of covering space is well-behaved with respect tothat of topological group. Indeed, let p : G→ H be a covering map between

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arbitrary topological spaces. If H is a topological group, then G inherits astructure of topological group that makes p a continuous homomorphism;this structure is unique once a lift of the identity element of H has beenchosen.Conversely, if G is a topological group and Γ is a discrete subgroup of itscentre, endowing G/Γ with the quotient topology we obtain a topologicalgroup and the covering map p : G→ G/Γ is a continuous homomorphism.

An open subgroup of a topological group is always closed as well (since allits cosets must be open as well). As a consequence, a connected topologicalgroup is always generated by any neighbourhood of its identity element.Moreover, a locally connected group always has exactly one connected, opensubgroup and it is the connected component of the identity.The following lemma will prove useful in the following chapters.

Lemma 1.1.4. Let G,H be locally compact, Hausdorff, second-countabletopological groups and let φ : G → H be a continuous homomorphism. If φis surjective, it is also open.

Proof. Let g ∈ G and let V be one of its neighbourhoods; we have to provethat φ(g) is an inner point of φ(V ). Without loss of generality we can assumethat g = e.Let U be a compact neighbourhood of e in G with the property that U−1U ⊆V . We can find countably many elements gn of G so that the sets gnU coverall G (since G is metrisable by the Nagata-Smirnov Theorem and henceLindelöf).Then H is covered by the compact sets φ(gnU) which cannot all be nowheredense by the Baire Theorem; hence some φ(gnU) must have an inner pointand the same holds for φ(U). Let φ(u) be an inner point of φ(U); then e isan inner point of φ(u−1U). Since u−1U ⊆ V , the statement follows.

Finally, we recall the following well-known result, which we will exploita huge number of times.

Theorem 1.1.5. Let G be a locally compact, Hausdorff topological group.Then, up to scalar multiplication, there exists a unique Borel measure µ onG such that:

1. µ is finite on compact sets and positive on open sets;

2. (Lg)∗µ = µ for each g ∈ G.

An analogous result holds if we replace left translations with right ones.

We will refer to µ as a left (right) Haar measure on G. We would alsolike to stress that by the term “Borel measure” we will always simply meana positive measure defined on all Borel sets.

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Definition 1.1.6. A locally compact, Hausdorff topological group G is saidto be unimodular if its left Haar measures are invariant also under righttranslations.

Lemma 1.1.7. Compact groups are unimodular.

Proof. Let µ be a left Haar measure for the compact group G. Since leftand right translations commute, also (Rg)∗µ is a left Haar measure, for eachg ∈ G. By the uniqueness part of Theorem 1.1.5, there exists a real numberλ(g) such that

(Rg)∗µ = λ(g) · µ .

Evaluating on G, we obtain:

0 < µ(G) = ((Rg)∗µ) (G) = λ(g) · µ(G) < +∞ ,

so it must be λ(g) = 1 for each g ∈ G. This means precisely that µ is a rightHaar measure as well.

We will see other examples of unimodular groups in Section 4.5.

1.2 Lie groups and Lie algebras

Definition 1.2.1. A Lie algebra is a vector space g over a field K = R,C,which is also equipped with a bilinear symmetric map [·, ·] : g× g→ g satis-fying the Jacobi identity:

[[X,Y ], Z] + [[Y,Z], X] + [[Z,X], Y ] = 0, ∀X,Y, Z ∈ g.

A morphism of Lie algebras is a bracket-preserving, linear map.

Given a vector space V overK, we can endow the space of endomorphismsof V with the usual commutator [f, g] = f g− g f , thus obtaining the Liealgebra gl(V ).For each element X ∈ g we can consider the map

adX : g→ g

Y 7→ [X,Y ]

and consequently define a morphism of Lie algebras

adg : g→ gl(g)

X 7→ adX

known as the adjoint representation of g. Its kernel is called the centre of g.

Definition 1.2.2. An ideal h of g is a vector subspace such that adX mapsg into h whenever X is in h.

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The image adg(g) of the adjoint representation of g is always an ideal ofgl(g). The Killing form of g is the symmetric, bilinear map κg : g × g → Rdefined by κg(X,Y ) := tr(adX adY ); we will simply denote it by κ whenthere is no ambiguity.

Definition 1.2.3. The Lie algebra g is said to be semisimple when its Killingform is non-degenerate.

A semisimple Lie algebra cannot have any nonzero abelian ideals. Inparticular adg provides an embedding g in gl(g).

Definition 1.2.4. A Lie group is a smooth manifold G which also has agroup structure, with the requirement that the map

G×G→ G

(x, y)→ x−1y

be smooth. A morphism of Lie groups is simply a smooth homomorphism.

If G is a Lie group, the homeomorphisms Lg, Rg, Cg are actually diffeo-morphisms for each g ∈ G.The tangent space of G at the identity element e can be endowed with a Liealgebra structure in the following way. Given two tangent vectors X,Y , it ispossible to extend them to G-invariant vector fields X, Y on G, i.e. vectorfields that are preserved by all the differentials dLg, g ∈ G. We then define

[X,Y ] := [X, Y ]e ,

where the second bracket represents a Lie bracket. We will simply call thisLie algebra the Lie algebra of G. The group G is said to be semisimple if itsLie algebra is.The following standard results clarify the relationship between Lie groupsand Lie algebras. They are Theorems 20.19 and 20.21 in [Lee12].

Theorem 1.2.5. Given Lie groups G,H with Lie algebras g, h and a mor-phism φ : G→ H, the differential def : g→ h is a morphism of Lie algebras.If G is simply connected, every morphism g→ h is the differential at e of aunique morphism G→ H.

Theorem 1.2.6. Every finite-dimensional real Lie algebra is isomorphic tothe Lie algebra of a simply connected Lie group, which is unique up to iso-morphism. In particular, connected Lie groups with isomorphic Lie algebrashave isomorphic universal covering spaces.

A Lie subgroup of G is a (not necessarily injective) immersion i : H → Gwhich is also a group homomorphism; with an abuse, we will also call “Liesubgroup” the image i(H). A Lie subgroup is said to be regular if the

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immersion is actually an embedding; some authors limit the definition of Liesubgroup to this case.As an example, the torus has several one-dimensional Lie subgroups; theregular ones are embedded circles, whereas the non-regular ones are injectiveimmersions of R with dense image.When i : H → G is a Lie subgroup, the image of the differential of i at theidentity element is always a subalgebra of the Lie algebra g of G. Conversely,each subalgebra h ⊆ g is the image of dei for some Lie subgroup i : H → G.The following is Corollary 20.13 in [Lee12].

Theorem 1.2.7. Regular Lie subgroups are precisely the closed (algebraic)subgroups.

IfG is a Lie group with Lie algebra g, it is possible to define an exponentialmap

exp: g→ G

in the following way: given X ∈ g and X the corresponding G-invariant fieldon G, we can consider the associated vector flow ΦX : G×R→ G and defineexpX as ΦX(e, 1) (the flow is defined on all of G×R because X has constantnorm if we endow G with a G-invariant Riemannian metric and any suchmetric is complete; see Lemma 1.4.4 in the next sections).The exponential map is smooth and its differential at 0 is the identity ofg. Moreover, exp sends one-dimensional subspaces of g to one-parametersubgroups of G (i.e. Lie subgroups of the form R→ G) and all one-parametersubgroups of G are obtained this way.For each vector space V , the group GL(V ) is a Lie group with Lie algebragl(V ). Furthermore, exp: gl(V ) → GL(V ) is the usual matrix exponential.Whenever H < GL(V ) is a Lie subgroup, its Lie algebra h ⊆ gl(V ) can berecovered thanks to the relation

h = X ∈ gl(V ) | exp(tX) ∈ H, ∀t ∈ R .

Theorem 1.2.8. A continuous homomorphism φ : G → H between Liegroups is always smooth.

Proof. The graph F of φ is a closed subgroup of G×H. Its Lie algebra is

f = (X,Y ) ∈ g⊕ h | φ(expG(tX)) = expH(tY ), ∀t ∈ R,

where g and h are the Lie algebras of G and H.Let πG, πH be the projections of G × H onto its factors. Since πG|F is aninjective homomorphism, if we prove that (deπG)|f is an isomorphism, thenit is easy to see that πG|F is a diffeomorphism of F onto G and the mapφ = πH (πG|F )−1 must be smooth. The injectivity of (deπG)|f is obvious,so we simply have to show that for each X ∈ g there exists some Y ∈ h such

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that (X,Y ) ∈ f.Take a neighbourhood U of the origin in g that is mapped diffeomorphicallyby the exponential map onto some neighbourhood U ′ of the identity elementof G; define similarly V ⊆ h and V ′ ⊆ H. If U, V are small enough, expG×Hmaps (U × V ) ∩ f diffeomorphically onto (U ′ × V ′) ∩ F ; moreover, we canassume that φ(U ′) ⊆ V ′. Now, for X ∈ g, there exists an integer n ∈ N sothat 1

nX ∈ U ; then, there must exist Y ∈ V with φ(expG( 1nX)) = expH Y

and a Z ∈ f with expG×H Z = (expG( 1nX), expH Y ). So Z = ( 1

nX,Y ) andnZ = (X,nY ) ∈ f.

If g is a Lie algebra, we can consider the subgroup Aut(g) < GL(g)of linear automorphisms preserving the bracket. Since Aut(g) is closed, itadmits a Lie group structure making it a regular Lie subgroup of GL(g).An endomorphism f of g is said to be a derivation if

f([X,Y ]) = [f(X), Y ] + [X, f(Y )]

holds for all X,Y ∈ g. The set Der(g) of all derivations of g is a subalgebraof gl(g) containing adg(g). A derivation is said to be inner if it is of the formadX for some X ∈ g.It is easy to check that the Lie algebra of Aut(g) is precisely Der(g). Wedefine the group Int(g) of inner automorphisms of g as the connected Lie sub-group of Aut(g) associated with the Lie algebra adg(g) of inner derivations.Note that this subgroup might not be regular in general.

Lemma 1.2.9. If g is semisimple, we have adg(g) = Der(g). In particularDer(g) is semisimple.

The adjoint representation of G is the morphism of Lie groups

AdG : G→ GL(g),

where AdG(g) is the differential at e of the map

Cg : G→ G

x 7→ gxg−1 .

The adjoint representations of g and G fit in the commutative diagram

GAdG // GL(g)

gadg //

exp

OO

gl(g)

exp

OO

and it is easy to check that AdG(G) = Int(g).

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Lemma 1.2.10. Let G a be a Lie group and Z its centre (which is a closedsubgroup). Then the Lie algebra z of Z is precisely the centre of the Liealgebra g of G.

Proof. The centre of g is contained in z. Indeed, if X,Y ∈ g satisfy [X,Y ] =0, then expX, expY will commute too (as is easy to check by observing thatthe G-invariant fields X, Y must have trivial Lie bracket at each point).Conversely, take X ∈ z; then exp(tX) lies in Z for each t ∈ R. In particularAdG(exp(tX)) = exp(adg(tX)) must be the identity of g and, deriving in t,we obtain adgX = 0.

Corollary 1.2.11. A connected Lie group is abelian if and only if its Liealgebra is abelian. In particular it must be of the form Rn × (S1)m.

Proof. The first part follows from the previous lemma. Let G be a connectedabelian Lie group with Lie algebra g. The simply connected Lie group withLie algebra gmust be Rk, with k = dim g, and it must cover G. So G = Rk/Γfor some lattice Γ ⊆ Rk. It is well-known that a lattice in Rk is alwaysisomorphic to some Zm, with m ≤ k, and that it admits a basis over Zwhich can be extended to a basis of Rk (over R). The result follows.

1.3 Cartan decompositions

Definition 1.3.1. A real Lie algebra g is said to be compact if the Lie groupInt(g) is compact.A subalgebra h ⊆ g is said to be compactly embedded if the connected Liesubgroup of Aut(g) associated with adg(h) ⊆ gl(g) is compact.

Remark 1.3.2. Let i : H → G be a Lie subgroup; even assuming that i isinjective, the topologies of H and i(H) are, in general, different (e.g. thetorus has injective, dense one-parameter subgroups).Still, there is no ambiguity in the notion of compact Lie subgroup. Sureenough, H is compact if and only if i(H) is compact (if i(H) is compact,then H is a closed, and hence regular, Lie subgroup of G and i must be anembedding; the other implication is trivial).In particular, the notion of compactly embedded subalgebra is well-defined.

Given a vector space V , a bilinear form (·, ·) and an endomorphism A,we say that (·, ·) is A-invariant if

(Av,w) + (v,Aw) = 0

holds for all v, w ∈ V . This condition is easily seen to be equivalent to

(etAv, etAw) = (v, w), ∀t ∈ R, ∀v, w ∈ V,

where e : gl(V ) → GL(V ) denotes the matrix exponential. Indeed, one cancheck that the exponential of an antisymmetric matrix is always orthogonal.

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Lemma 1.3.3. Let H be a compact subgroup of GL(V ) for some finite-dimensional real vector space V . Then there exists a scalar product (·, ·) onV that is preserved by all elements of H.

Proof. Let µ be a left Haar measure on H; clearly µ is finite. Having fixedan arbitrary scalar product (·, ·)0, we can define

(v, w) :=

∫H

(h(v), h(w))0 dµ(h).

The fact that this scalar product is preserved by all elements of H followsfrom the invariance under left translations of µ.

Lemma 1.3.4. A compact real Lie algebra g always admits a scalar productwhich is g-invariant, i.e. adX-invariant for all X ∈ g.

Proof. The group Int(g) is compact, so the previous lemma provides anInt(g)-invariant, and hence g-invariant, scalar product on g.

We remark that by “scalar product” we mean a positive definite, sym-metric bilinear form.

Lemma 1.3.5. If a subalgebra h ⊆ g is compactly embedded, then thereexists an h-invariant scalar product on g.Moreover, a compactly embedded subalgebra is always compact.

Proof. Let Intg(h) be the connected subgroup of Int(g) with Lie algebraadg(h). If h is compactly embedded, the group Intg(h) is compact andLemma 1.3.3 yields an Intg(h)-invariant, and hence adg(h)-invariant, scalarproduct on g.Let Auth(g) be the group of automorphisms of g that leave h invariant.Then the restriction homomorphism r : Auth(g)→ Aut(h) is continuous andit maps the compact group Intg(h) onto Int(h). Hence, Int(h) is compact.

Lemma 1.3.6. A semisimple real Lie algebra g is compact if and only if itsKilling form is negative definite.

Proof. If the Killing form κ is definite, the group O(κ) < GL(g) of linearautomorphisms preserving κ is compact. Since Aut(g) and Int(g) are closedsubgroups of O(κ), they are compact as well.Assume now that g is compact. We can put a g-invariant, scalar product ong; considering an orthonormal basis of g (as a vector space), all the elementsof adg(g) will be represented by antisymmetric matrices. In particular, ifadX is represented by the matrix A = (ai,j), then A 6= 0 since the centre ofg is trivial and we have:

κ(X,X) = tr(A2) = −∑i,j

a2i,j < 0 .

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Definition 1.3.7. A real form in a complex Lie algebra g is a real subalgebrag0 such that g = g0 ⊕ (i · g0).

For example, any real Lie algebra g0 is a real form of its complexificationg = g0 ⊗ C.To each real form we can associate an R-linear automorphism σ : g → g bysetting σ|g0 = id and σ|i·g0 = −id; with this definition, it is also clear thatσ preserves the bracket and σ2 = id. We shall say that σ is the conjugationrelative to g0. Conversely, the set of fixed points of any R-linear involutionanticommuting with the multiplication by i and preserving the bracket is areal form.In the setting above, the two Lie algebras share the same Killing form (moreprecisely: κg0 = κg|g0); hence, g is semisimple if and only if g0 is.The following is a consequence of the existence of Chevalley generators forsemisimple complex Lie algebras. See [Hel78] or [Hum72] for more details.

Theorem 1.3.8. Every semisimple complex Lie algebra contains a real formthat is compact.

Two real forms g1, g2 of g are said to be compatible if, given σ1, σ2 theirconjugations, g1 is σ2-invariant and vice versa (though the latter invariancealways follows from the former). If this is the case, we obtain a decomposition

g1 = (g1 ∩ g2)⊕ (g1 ∩ (i · g2)) .

Theorem 1.3.9. Every real form g0 of a semisimple complex Lie algebra gis compatible with some compact real form.

Proof. Let u be a compact real form of g, τ its conjugation and σ the con-jugation with respect to g0. Let κ be the Killing form of g.We are looking for some f ∈ Aut(g) such that the compact form f · u iscompatible with g0; equivalently, we desire that σ and f τ f−1 commute.Since κ|u = κu is real-valued and τ(iX) = −i · τX for all X ∈ g, it is easy tocheck that κ(τX, τY ) = κ(X,Y ) for all X,Y ∈ g.Hence the R-bilinear form

κτ (X,Y ) := −κ(X, τY );

is clearly Hermitian on g. Moreover, κu being negative definite, it is easy tocheck that κτ is positive definite.Consider now the automorphism N = στ of g (it is C-linear and bracket-preserving since σ, τ are antilinear and bracket-preserving); it satisfies

κτ (N(X), Y ) = κτ (X,N(Y )), ∀X,Y ∈ g

and hence is represented by a diagonal matrix in some basis of g whichis κτ -orthonormal. Moreover, the eigenvalues of N are real, so that P =

15

N2 has only positive eigenvalues and can be written as P = eQ, for somediagonalisable operator Q with real eigenvalues.It is easy to check that etQ, t ∈ R is a one-parameter subgroup of Aut(g).Define τ ′ = exp(1

4Q) τ exp(−14Q). Since τNτ−1 = N−1, we obtain

τetQτ−1 = e−tQ for all t ∈ R and this easily implies στ ′ = τ ′σ after somestraightforward computations.

Remark 1.3.10. Compatible compact real forms u1, u2 ⊆ g must coincide.If this were not the case, the restriction of the Killing form of g to u1∩ (i ·u2)would have to be both positive and negative definite.

Corollary 1.3.11. All compact real forms in a semisimple complex Lie al-gebra are conjugated via inner automorphisms.

Proof. This follows immediately from the proof of Theorem 1.3.9 and theprevious remark.

We can now define the Cartan decomposition of a semisimple real Liealgebra, which will be essential in the study of symmetric spaces.

Definition 1.3.12. Let g be a semisimple real Lie algebra; g can be seen asa real form in the complex semisimple Lie algebra gC := g ⊗ C. Let u be acompact real form of gC which is compatible with g.A Cartan decomposition of g is a decomposition g = k ⊕ p, where k is thesubalgebra g ∩ u and p is the vector space g ∩ (i · u).

Lemma 1.3.13. Let g = k ⊕ p be a Cartan decomposition. Then k is amaximal compactly embedded subalgebra in g.

Proof. Let u be the compact real form of gC whose intersection with g is k.Let IntR(gC) be the group of inner automorphisms of gC viewed as a real Liealgebra. Let σ be the conjugation associated to the real form g of gC. Thegroup Int(g) can be viewed as the identity component of the closed subgroupof IntR(gC) given by those automorphisms that commute with σ; in partic-ular, we can identify Int(g) with a closed subgroup of IntR(gC). Similarly,Int(u) can be seen as a compact subgroup of IntR(gC).Then Int(u)∩ Int(g) is a compact subgroup of IntR(gC) (and of Int(g)) withLie algebra h; hence h is compactly embedded.If k were not maximal, there would be a bigger compactly embedded subal-gebra k′, which would then be forced to contain some X ∈ p \ 0.Being τX = −X, an easy computation shows that

κτ (adXY,Z) = κτ (Y, adXZ),

for all Y, Z ∈ g. Hence, adX is diagonalisable and nonzero (since semisimplealgebras have trivial centre). But then exp(t·adX), t ∈ R, is a one parametersubgroup of GL(g) that cannot be contained in any compact Lie subgroup,contradicting the fact that the Lie subgroup of Aut(g) associated with adg(k′)is compact.

16

Finally, Cartan decompositions are unique up to inner automorphisms,as shown by the next result.

Theorem 1.3.14. Let g be a real semisimple Lie algebra with Cartan de-compositions g = k1⊕p1 = k2⊕p2. Then there exists an inner automorphismof g conjugating k1, k2 and also p1, p2.

Proof. The two Cartan decompositions are associated with compact realforms u1, u2 of gC; let τ1, τ2 be the corresponding conjugations.Exactly as in the proof of Theorem 1.3.9, there exists a one-parameter sub-group etQ, t ∈ R of automorphisms of gC such that eQ = (τ1τ2)2 andexp(1

4Q) must send u2 to a compact real form that is compatible, and hencecoincident, with u1 (see Remark 1.3.10).We just have to check that the restriction of exp(1

4Q) to g0 is an inner au-tomorphism. Since both τ1, τ2 leave g0 invariant, so do τ1τ2 and each etQ.The restriction to g0 of this one-parameter subgroup of Aut(g) is thereforea one-parameter subgroup of Aut(g0) and has to lie in the identity compo-nent. But, by semisimplicity and Lemma 1.2.9, this component is preciselyInt(g0).

1.4 Some basics of Riemannian geometry

Let (M, g) be a connected Riemannian manifold. Throughout this disserta-tion we will denote by I(M) the group of isometries of M and by I+(M)the subgroup of orientation-preserving transformations. Moreover, I0(M)will be the identity component of I(M) (with respect to the compact-opentopology).The following is an easy but useful lemma.

Lemma 1.4.1. Let f, g ∈ I(M) and assume that, for some x ∈M , we havef(x) = g(x) and dxf = dxg. Then f = g.

Proof. The subset of TM where df and dg coincide must be open and closed.

We recall the following standard result originally due to S. B. Myers andN. E. Steenrod ([MS39]). An easier proof in the case of symmetric spacescan be found as Lemma 3.2 in Chapter IV of [Hel78].

Theorem 1.4.2. Let M be any Riemannian manifold. The group I(M) hasa Lie group structure which is compatible with the compact-open topology.Moreover the map

I(M)×M →M ×M(f, x) 7→ (f(x), x)

is proper and smooth with respect to this structure.

17

Definition 1.4.3. A Riemannian manifold (M, g) is said to be homogeneousif I(M) acts transitively on M .

Lemma 1.4.4. A homogeneous Riemannian manifold is complete.

Proof. Since the isometry group acts transitively, the injectivity radius isconstant over the manifold. In particular, all geodesics are defined on thewhole real line. The conclusion follows from the Hopf-Rinow Theorem.

Lemma 1.4.5. The isometry group of a connected, homogeneous Rieman-nian manifold M has finitely many connected components.

Proof. Let G = I(M) and Gp the stabiliser of some point p ∈ M . Thespace G/Gp with the quotient topology is clearly homeomorphic to M : thecontinuous bijection G/Gp →M induced by the action of G on p, is properbecause of Theorem 1.4.2 and hence a homeomorphism.So G/Gp is connected and Gp must have at least as many connected com-ponents as G; but Gp is compact, hence it has finitely many connectedcomponents.

We recall that, for a Riemannian manifold (M, g) with Levi-Civita con-nection ∇, the curvature tensor is the (1, 3)-tensor satisfying

R(X,Y )Z := (∇Y∇X −∇X∇Y −∇[Y,X])Z,

whenever X,Y, Z are vector fields on M . Equivalently, we can consider the(0, 4)-tensor

R(X,Y, Z,W ) := g(R(X,Y )Z,W ).

Given p ∈ M and a two-dimensional subspace V of TpM , the sectionalcurvature of V is

secV := R(u, v, u, v)

where (u, v) is any orthonormal basis of V . It is easy to check that this iswell-defined by exploiting the symmetries of the curvature tensor.The manifold M is said to have nonpositive curvature if secV ≤ 0 for eachV ⊆ TpM and each p ∈M .We will need the following celebrated results on nonpositively-curved mani-folds.

Theorem 1.4.6 (Cartan-Hadamard). Let (Mn, g) be a complete, connectedRiemannian manifold with nonpositive curvature and let p be one of itspoints. Then the exponential mapping

expp : TpM →M

is a smooth covering map. In particular, the universal covering space of Mis diffeomorphic to Rn.

18

Theorem 1.4.7 (Cartan). Let (M, g) be a simply connected, complete Rie-mannian manifold with nonpositive curvature and let G be any subgroup ofI(M). If G has a bounded orbit, then G fixes a point.

Remark 1.4.8. Let M be a simply connected, nonpositively-curved man-ifold. Given a (k + 1)-tuple of points (x0, ..., xk) ∈ Mk+1, we can define acanonical, smooth singular simplex ∆k → M which we will call the straightsimplex with vertices x0, ..., xk and which we will denote by ∆(x0, ..., xk).The definition is obtained via an inductive process and goes as follows.First, for k = 0, the simplex ∆(x) is simply the point x. Then, sup-pose to have defined all straight simplices with at most k vertices and letf0 : ∆k−1 →M be the simplex ∆(x1, ..., xk). View ∆k as the set

(λ0, ..., λk) | λi ≥ 0, λ0 + ...+ λk = 1 .

Define f : ∆k →M so that each path in ∆k of the form

t 7→ (1− t) · (λ0, 0, ..., 0) + t · (0, λ1, ..., λk)

is sent to the geodesic from x0 to f0(λ1, ..., λk) in M , parametrised by amultiple of arc-length. Well-definition and smoothnes of f follow from theCartan-Hadamard Theorem.We will say that ∆(x0, ..., xk) is the geodesic cone on the simplex ∆(x1, ..., xk)with vertex x0.Observe moreover that the faces of ∆(x0, ..., xk) are precisely the straightsimplices ∆(x0, ..., xj , ..., xk). However, in general, permuting the vertices of∆(x0, ..., xk) we will obtain different straight simplices.

Finally we recall the concept of totally geodesic submanifold, which weshall briefly need in Chapter 6.

Definition 1.4.9. A submanifold Σ of the Riemannian manifold M is saidto be totally geodesic if, for each p ∈ Σ, the Riemannian exponential map ofM takes a neighbourhood of the origin in TpΣ to a subset of Σ.

The following is Exercise 8 in Chapter 5 of [Pet06].

Proposition 1.4.10. Let M be a Riemannian manifold and Σ a totallygeodesic submanifold, which we consider with the induced Riemannian met-ric. Then the curvature tensor of Σ is simply the restriction of the curvaturetensor of M .

In particular, if V is a plane in a tangent space of Σ, the sectional cur-vature of V is the same whether we compute it in Σ or M .

19

1.5 Some functional analysis

In this section we collected a few basic facts on Fréchet spaces and on somerepresentations of topological groups that they allow us to construct. Theseresults will be used mainly in Chapter 3 when we shall define the continuouscohomology spaces of a group.In §1.5.3 we will also recall a few properties of the Bochner integral, a gen-eralisation of the Lebesgue integral to functions taking values into a Banachspace. We will need this construction in Chapter 4.

1.5.1 Fréchet spaces

Definition 1.5.1. A topological vector space is a Hausdorff topological spaceE which also has a vector space structure (over R or C) such that vectoraddition and scalar multiplication are continuous.In addition, we say that E is locally convex if the origin has a neighbourhoodbasis made of convex open sets.

We will always consider implicitly real topological vector spaces.

Definition 1.5.2. A locally convex topological vector space E is said tobe a Fréchet space if its topology is induced by a complete metric that isinvariant under translations.

In particular, all Banach spaces are Fréchet spaces. Throughout the restof this section, E will be a Fréchet space and d will be a translation-invariant,complete metric inducing its topology.We will exploit the following examples of Fréchet spaces in Chapter 3.

Example 1.5.3. Let X be a locally compact, Hausdorff, second-countabletopological space. The space of continuous functions from X to E, endowedwith the compact-open topology, is a Fréchet space; we will denote it byC(X,E).Indeed, we can construct a metric on C(X,E) in the following way. LetX =

⋃n∈NKn be an exhaustion of X by compact sets and define for maps

f, g ∈ C(X,E)

d(f, g) :=∑n≥0

1

2n+1·min

[(supx∈Kn

d(f(x), g(x))

), 1

].

It is easy to check that this defines a complete, translation-invariant metricand that E is a locally convex topological vector space with the associatedtopology.Convergence with respect to d is precisely uniform convergence on compactsets (see e.g. Theorem 46.8 in [Mun00]).

20

Example 1.5.4. Let Mn be a smooth manifold. Also the space C∞(M,E)of smooth functions from E toM can be endowed with a structure of Fréchetspace.We will show it only for E = R, since this is the only case that we will need;however the general case is very similar.Cover M with countably many compact sets Kα, α ∈ A, each endowed withcoordinates xα1 , ..., xαn (i.e. each Kα is contained in a chart). Given α ∈ Aand β1, ..., βn ∈ N, we can define

dα,β1,...,βn(f, g) := supx∈Kα

∣∣∣∣∣ ∂β1+...+βn

∂xα,β11 ...∂xα,βnn

f(x)− ∂β1+...+βn

∂xα,β11 ...∂xα,βnn

g(x)

∣∣∣∣∣,for f, g in C∞(M,R). Then order all possibles (n + 1)-uples (α, β1, ..., βn),associating an integer i to each of them, and define:

d∞(f, g) :=∑i∈N

1

2i+1·min

(di(f, g), 1

).

Again it is easy to check that all the required properties are satisfied. Inparticular, completeness follows from standard results on the convergence ofderivatives.The notion of convergence defined by d∞ coincides with that of uniformconvergence on compact sets of f and all its partial derivatives. Hence, thetopology induced on C∞(M,R) by d does not depend on the choices involvedin the definition of the metric.

Let now F be a closed subspace of E. The vector space E/F becomes atopological vector space when we endow it with the quotient topology; theprojection E → E/F is linear, continuous and open. Moreover, it is clearthat E/F is locally convex if E is.

Lemma 1.5.5. The quotient E/F of a Fréchet space by a closed subspace isa Fréchet space as well.

Proof. If d is a complete, translation-invariant metric inducing the topologyof E, we can define on E/F :

d(e1, e2) := d(e1 − e2, F ).

It is immediate to check that d is well-defined and translation-invariant.Moreover, given e1, e2, e3 ∈ E:

d(e1 − e2, x) + d(e2 − e3, y) ≥ d(e1 − e3, x+ y) ≥ d(e1 − e3, F ), ∀x, y ∈ F,

so that d(e1, e2) + d(e2, e3) ≥ d(e1, e3) and d is a metric. Since the balls ofd are precisely the projections of the balls of d, the distance d induces the

21

topology of E/F .Now we just have to check completeness. Let (en)n∈N be a Cauchy sequencein E/F ; up to passing to a subsequence, we can assume that d(en, en+1) <1/2n. Choosing appropriately each en ∈ E, we can obtain d(en, en+1) < 1/2n

as well. Since (E, d) is complete, (en) converges and so does (en).

Definition 1.5.6. Let E,F be topological vector spaces and f : E → F acontinuous linear map. If f is injective, we say that it is strong if it admitsa left-inverse which is linear and continuous.In the general case, f is said to be strong if both the inclusion ι : Ker f → Eand the quotient map f : E/Ker f → F are.

The previous definition is the correct notion of strong map in the categoryof topological vector spaces and we will refer to it repeatedly in Chapter 3.However, in Chapter 4 we will meet a slightly different notion of strong map,which will be appropriate in the category of Banach spaces. It will be alwaysclear from the context, which notion of strength we are adopting.

Lemma 1.5.7. Let f : E → F be a strong, continuous, linear map betweentopological vector spaces. Then there exist closed subspaces E1 ⊆ E andF1 ⊆ F with E = Ker f ⊕E1 and F = Im f ⊕F1. Moreover, the restrictionof f to E1 is a homeomorphism with Im f .

Proof. Let α, β be left inverses for ι and f , respectively. Then E1 = Ker αand F1 = Ker β satisfy the first part of our statement.It is also evident that the restriction of f to E1 is a continuous bijection withIm f . Let π the quotient projection from E to E/Ker f ; clearly π|E1 is ahomeomorphism. Hence (π|E1)−1 β is a continuous inverse for f .

Proposition 1.5.8. Let E be a topological vector space and F a finite-dimensional subspace. Then F is closed.

Proof. This is 1.21 in [Rud91].

Theorem 1.5.9. Let f : E → F be a continuous, surjective linear mapbetween Fréchet spaces. Then f is open.

Proof. This follows from 2.11 in [Rud91] and the Baire Theorem. The proofis identical to the one usually given for Banach spaces.

Corollary 1.5.10. Let f : E → F be a continuous, linear map betweenFréchet spaces. If G = Imf has finite codimension in F , then it is a closedsubspace.

Proof. Let V be a finite-dimensional subspace of F such that F = G ⊕ V ;the subspace V is closed by Proposition 1.5.8.The space E/Kerf is Fréchet by Lemma 1.5.5 and so must be (E/Kerf)×V

22

with the product topology.We can define a continuous, linear bijection (E/Kerf)× V → F as f on thefirst summand and as the inclusion of V in F on the second summand. Thishas to be a homeomorphism by Theorem 1.5.9. Since E/Kerf is closed in(E/Kerf)× V , this means that G is closed in F .

1.5.2 Fréchet G-modules

Let G be a topological group.

Definition 1.5.11. A Fréchet G-module is a Fréchet space E provided witha left action of G on E by linear homeomorphisms such that the map

G× E → E

(g, e) 7→ g · e

is continuous. We will also call this a continuous representation of G.

Definition 1.5.12. A morphism of G-modules (or G-map) is a linear con-tinuous map f : E → F conjugating the actions of G, i.e satisfying

f(g · e) = g · (f(e))

for each g ∈ G and e ∈ E.

Definition 1.5.13. The subspace of invariants of a Fréchet G-module E is

EG := e ∈ E | g · e = e, ∀g ∈ G.

Observe that the subspace of invariants is always a closed subspace ofthe G-module E; in particular, it inherits the Fréchet structure.The following result will prove particularly useful for constructingG-modules;it can be found (in the more general context of barrelled spaces) in [Bou04]as Proposition 1 in Chapter VIII, §2, No.1.

Proposition 1.5.14. Let G be a locally compact, Hausdorff group acting ona Fréchet space E by linear homeomorphisms. Suppose that this representa-tion of G is separately continuous, i.e. for each e ∈ E the map

ve : G→ E

g → g · e

is continuous. Then this is actually a continuous representation of G and Eis a G-module with this action.

We will always assume G to be Hausdorff and locally compact in thefollowing discussion.

23

Example 1.5.15. Let E be a Fréchet G-module and X a locally compact,Hausdorff, second-countable topological space. Suppose that G acts on theleft on X and that the map

G×M →M

(g, x)→ g · x

is continuous. Then G acts on the left also on the Fréchet space C(X,E)defined in Example 1.5.3 by

(g · f)(x) := g ·(f(g−1 · x)

).

We obtain a Fréchet G-module that we will keep denoting by C(X,E).Indeed, thanks to Proposition 1.5.14, we only have to check that this repre-sentation of G is separately continuous and that G acts by homeomorphisms.To prove the former assertion, assume that there exists a sequence gn → gin G and some f ∈ C(X,E) such that gn · f 6→ g · f . Then we would be ableto find some compact set K ⊆ X where (gn ·f) does not converge uniformly;so we could find some ε > 0 and points xn ∈ K with

d(gn · (f(g−1n xn)), g · (f(g−1xn))) ≥ ε, ∀n ∈ N.

Without loss of generality, we can assume that xn → x for some x ∈ K (sinceX is metrisable by the Nagata-Smirnov Theorem); but then both sequencesin the previous inequality would converge to the point g · (f(g−1x)), whichis absurd.To prove the second assertion, we only have to show that, if g ∈ G andfn → f uniformly on compact sets, then g · fn → g · f as well. TakeK ⊆ X compact and observe that the union K of all fn(K) and f(K) mustbe compact in E; hence, g|K is uniformly continuous, which implies ourstatement.

We will be especially interested in the G-modules C(Gn, E) in the laterchapters.

Example 1.5.16. Let G be a Lie group acting on the left on a smoothmanifold M , so that the map

G×M →M

(g, x)→ g · x := τ(g)(x)

is smooth. Then also the space C∞(M,E) defined in Example 1.5.4 is aFréchet G-module, with the restriction of the action of G on C(M,E).In the case where E = R with the trivial action of G, whis is particularlyeasy to prove. One just has to proceed as in the previous example, with theadditional observation that, given f ∈ C∞(M,E), each partial derivative ofg · f can be expressed as a sum of products of partial derivatives of τ(g−1)and partial derivatives of f , each of them then composed with τ(g−1).

24

Example 1.5.17. In the setting of the previous example, we can also definea left action of G on the space of p-forms Ωp(M,R) by

g · ω := (τ(g−1))∗ω .

We can embed Ωp(M,R) in C∞(ΛpTM,R) in an obvious way. If we equipthe space C∞(ΛpTM,R) with the topology defined in Example 1.5.4, theimage of this embedding is closed since it consists precisely of those mapsthat are linear on each fibre of ΛpTM →M . Therefore, endowing Ωp(M,R)with the subspace topology induced by this embedding, we obtain a Fréchetspace.The action of G on Ωp(M,R) is a continuous representation, since it is simplythe restriction of the action ofG on ΛpTM given in Example 1.5.16 (an actionof G on ΛpTM is induced canonically by the action of G on M).Finally, recall that the differential dp : Ωp(M,R) → Ωp+1(M,R), is definedas

(dpω)q [X0, ..., Xp] =:

p∑j=0

(−1)j ·Xj

(ωq

[X0, ..., Xj , ..., Xp

])+

+∑i<j

(−1)i+j · ωq[[Xi, Xj ], X0, ..., Xi, ..., Xj , ..., Xp

],

where X0, ..., Xp are vector fields near q ∈M .It is easy to check that dp is continuous with respect to our topologies.

Example 1.5.18. LetE,F be FréchetG-modules, with F finite-dimensional,and let d be a complete, translation-invariant metric inducing the topologyof E.The vector space E ⊗ F has a structure of Fréchet space given by viewingit as a direct sum of a finite number of isometric copies of (E, d) and byconsidering the product metric.The group G acts on E ⊗F by g · (e⊗ f) := (g · e)⊗ (g · f); this way E ⊗Fbecomes a Fréchet G-module.

1.5.3 Integration of vector-valued functions

We will need the following generalisation of the Lebesgue integral. See[Mon01] and [JU77].

Theorem 1.5.19. Let X be a metrisable, second-countable topological space,E a Banach space and f : X → E a continuous function. Let µ be a σ-finiteBorel measure on X.We say that f is Bochner integrable if∫

X‖f(x)‖ dµ <∞

25

and in this case it is possible to define a (unique) element∫X f(x) dµ of E,

called Bochner integral, such that the result is the usual Lebesgue integral ifE is finite-dimensional and furthermore, for a general E and for each linear,continuous operator to another Banach space T : E → F , we have

T

(∫Xf(x) dµ

)=

∫XT (f(x)) dµ.

Moreover:

1. ‖∫X f(x) dµ‖ ≤

∫X ‖f(x)‖ dµ;

2.∫X(αf + βg) = α

∫X f + β

∫X g, whenever α, β ∈ R and g ∈ C(X,E);

3. the Fubini-Tonelli Theorem holds also for the Bochner integral;

4. if µ is finite, Y is a topological space and F : X×Y → E is continuousand bounded, then the map F : Y → E defined by

F (y) :=

∫XF (x, y) dµ(x)

is continuous.

26

Chapter 2

Symmetric Spaces

This chapter is devoted to the study of symmetric spaces. In Section 2.1we will see how this geometric concept can be rephrased in purely algebraicterms. Geodesic symmetries correspond to involutions of Lie groups and Liealgebras and the Riemannian exponential map can be expressed in terms ofthe exponential map of a Lie group.Then we will show that symmetric spaces decompose into factors of threedifferent types, each with a clear behaviour in terms of sectional curvatureand compactness. It will also become clear that we are mainly interested insemisimple Lie groups.Finally, in Section 2.4 we will show that semisimple groups always containmaximal compact subgroups and originate symmetric spaces.

2.1 Globally symmetric spaces

Let (Mn, g) be a Riemannian manifold. An isometry f of M fixing a pointx is called the geodesic symmetry at x if dxf = −id.By Lemma 1.4.1, all geodesic symmetries are involutions. Moreover, x isalways an isolated fixed point for the geodesic symmetry at x.

Definition 2.1.1. A connected Riemannian manifold M is said to be aglobally symmetric space if there is a geodesic symmetry sx at x for everyx ∈M .A locally symmetric space is a Riemannian manifold covered by a globallysymmetric space. We will usually refer to globally symmetric spaces simplyas symmetric spaces.

Proposition 2.1.2. Symmetric spaces are homogeneous.

Proof. We start by proving completeness. Assume that (a, b) ⊆ R is themaximal domain of some geodesic γ in the symmetric space M . If b < ∞,choose some a+b

2 < t < b and consider the geodesic arc γ′ = sγ(t) γ.

27

A portion of γ′ can be glued to γ, yielding an extension of its domain to(a, 2t − a), since 2t − a > b. This violates the maximality of (a, b), forcingus to conclude that b = +∞ and, similarly, a = −∞. The Hopf-RinowTheorem then guarantees completeness.Now take x, y ∈ M and a geodesic arc γx,y connecting them; its existencefollows again from the Hopf-Rinow Theorem. If p is the midpoint of γx,y, itis immediate that sp exchanges x and y.

The presence of geodesic symmetries allows us to express the paralleltransport in terms of one-parameter subgroups of the isometry group. Wewill see later in this section (Lemma 2.1.6) that even the actual geodesicsare given by the action of one-parameter subgroups of the isometry group.

Proposition 2.1.3. Let M be a symmetric space, p one of its points and γa geodesic with γ(0) = p. Call st the geodesic symmetry at γ(t) and definethe isometries Tt := st/2s0. Then:

1. d0Tt is the parallel transport from p to γ(t) along γ;

2. the map t 7→ Tt is a one-parameter subgroup of I(M).

Proof. 1. Take a tangent vector v at γ(t/2) and let v0 and vt be thetangent vectors at p and γ(t) obtained by parallel transport. Sincedst/2(v) = −v and isometries respect the parallel transport, it is clearthat dst/2(v0) = −vt. Hence dTt(v0) = vt, that is dpTt is the paralleltransport along γ.

2. First, it is easy to check that TtTt′ = Tt+t′ for each t, t′ ∈ R; indeed thisis equivalent to st/2s0st′/2 = s(t+t′)/2 and this equality follows from thefact that both sides of it are isometries fixing γ((t+ t′)/2) and actingas −id on the tangent space at that point (which is a consequence ofthe description of the differentials of geodesic symmetries given in thefirst part of the proof).Hence we have obtained a homomorphism t 7→ Tt of R into I(M). Itis easy to check that this map is continuous so that we can appeal toTheorem 1.2.8 to prove that it is actually smooth.

A proof of the following easy proposition can be found in [Hel78] (seeTheorem 4.2 and Proposition 4.3 in Chapter II).

Proposition 2.1.4. • Let G be a Lie group and H a closed subgroup.Then the space of left cosets G/H admits a unique smooth structure(compatible with the quotient topology) such that the projection

π : G→ G/H

is a smooth submersion.

28

• Let G act transitively on a smooth manifold M so that the map

G×M →M ×M(g, x) 7→ (g · x, x)

is smooth and proper. Then, denoting by Gp the set of all elements ofG fixing some p ∈M , the map

πp : G/Gp →M

g 7→ g · p

is a diffeomorphism.

Theorem 1.4.2 and Proposition 2.1.4 clearly imply that any connected,homogeneous Riemannian manifold M is diffeomorphic to G/H where G issome connected Lie group and H is a compact subgroup of G.Sure enough, it suffices to set G = I0(M) and to choose some p ∈M ; settingH = Gp we obtain the desired diffeomorphism πp : G/H →M .

Remark 2.1.5. When M is a symmetric space, we can say more. Thegeodesic symmetry sp is an isometry and so it induces by conjugation anorder-two automorphism σp of I(M) and hence of G. Since σp commuteswith all isometries fixing p, we have σp|H = id and the diagram

G/Hπp //

σp

M

sp

G/H

πp //M

clearly commutes.Furthermore, deσp is an involutive automorphism of the Lie algebra g of G;its set of fixed points is precisely the Lie algebra h of H. Indeed, it is clearthat deσp|h is the identity and, if deσp were to fix some other vector in g,we would get a one-parameter subgroup of G acting nontrivially on p andcommuting with sp; in particular sp would fix pointwise the orbit of p underthe action of this subgroup and p would not be isolated in Fix(sp).

Lemma 2.1.6. Mantaining the notation of the previous remark, let p be theeigenspace of deσp relative to the eigenvalue −1. Then the curves

γX(t) := πp(exp(tX)),

with X ∈ p, are precisely the geodesics in M through p.

Proof. Choose a geodesic γ in M with γ(0) = p, call st the geodesic symmetryin γ(t) and define Tt = st/2s0. By Proposition 2.1.3, Tt is a one-parameter

29

subgroup of G; in particular there exists X ∈ g with Tt = exp(tX), ∀t ∈ R.Observe that

σp(Tt) = s0Tts0 = s0st/2 = T−1t = T−t,

so that it must be deσp(X) = −X, i.e. X lies in p. It is immediate to checkthat γ = γX .

Conversely, if G is a connected Lie group and H is a compact subgroup,we can always endow the smooth manifold G/H with a Riemannian metricthat turns it into a homogeneous Riemannian manifold, as we will now show.Every g ∈ G induces a diffeomorphism

τ(g) : G/H → G/H

xH → gxH

and the elements of G fixing the preferred point o = H ∈ G/H are preciselythe elements of H. Via the map τ , the group G acts transitively on themanifold M = G/H.Denote by g, h the Lie algebras of G,H respectively. We can endow g withan AdG(H)-invariant scalar product because of Lemma 1.3.3. So we candecompose g = h⊕h⊥, where both h and h⊥ are AdG(H)-invariant subspaces.The differential of π : G→ G/H at e is zero on h, while it identifies h⊥ andToM via an isomorphism. In particular, ToM inherits the AdG(H)-invariantmetric that we have on h⊥. This metric is preserved by doτ(h) wheneverh ∈ H since the diagram

G

π

Ch // G

π

G/H //τ(h) // G/H

commutes. As a consequence, we can transport this scalar product to eachpoint of M using some τ(g) and obtain a well-defined, homogeneous, Rie-mannian metric on M such that τ(g) is an isometry for each g ∈ G.Observe that only the compactness of AdG(H) (and not that of H) wasneeded for this construction.

Remark 2.1.7. If, in addition to our pair of Lie groups (G,H), we havean involution σ of G so that Fix(deσ) = h and σ|H = id, we can endowM = G/H with a symmetric-space structure.Since deσ is an involution of g whose 1-eigenspace is h, we can decomposeg = h ⊕ p, where p is the eigenspace of deσ associated with the eigenvalue−1. Since h and p are both AdG(H)-invariant, we can obtain as above anAdG(H)-invariant metric on g where p = h⊥ and this defines a structure ofhomogeneous Riemannian manifold on M with τ : G→ I(M).We now wish to prove that M is a symmetric space with this metric; to do

30

so, it suffices to check the existence of a geodesic symmetry at o. But σ isthe identity on H and hence descends to a map s : M → M which fixes o;the fact that dos = −id follows immediately from the observation that deπidentifies p with ToM .Observe that the geodesic symmetry at o (and hence at all points) does notdepend on our choice of the scalar product on g.

Lemma 2.1.8. In the setting of the previous remark, the geodesics througho in G/H are precisely the curves

γX(t) = π(exp(tX)),

with X ∈ p. In particular the geodesics through o do not depend on thechosen metric on G/H.

Proof. Clearly we have a continuous homomorphism

τ : G→ I(G/H)

g 7→ τ(g) .

Theorem 1.2.8 guarantees that τ is smooth. Note that the diagram

I(G/H)πo // G/H

G

τ

OO

π // G/H

id

OO

is commutative. Moreover, if σo is the involution of I(G/H) constructed inRemark 2.1.5, it is easy to check that τσ = σoτ ; indeed, this is equivalentto τ(σ(g)) = soτ(g)so holding for each g ∈ G and this is clear from thedefinition of σ.We can decompose the Lie algebra g of I(G/H) as g = h⊕ p, where h, p arethe eigenspaces of σo. Since τσ = σoτ , it is clear that deτ maps p into p.Actually, we see that it must be deτ(p) = p, as π, π map p, p isomorphicallyonto To(G/H).We conclude by invoking Lemma 2.1.6.

Remarks 2.1.5 and 2.1.7 lead us to the following definition.

Definition 2.1.9. A Riemannian symmetric pair is a pair of Lie groups(G,H), where G is connected and H is a closed subgroup of G such that:

• AdG(H) is compact;

• there exists an involution σ of G satisfying σ|H = id and Fix(deσ) = h,where h is the Lie algebra of H.

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The previous discussion shows that a symmetric space always defines aRiemannian symmetric pair (Remark 2.1.5) and a Riemannian symmetricpair always defines a symmetric space (Remark 2.1.7).We can also give a slightly different notion involving only Lie algebras.

Definition 2.1.10. An orthogonal symmetric Lie algebra is a real Lie al-gebra g equipped with an involutive automorphism s, such that Fix(s) is acompactly embedded subalgebra.If Fix(s) intersects trivially the centre of g, we speak of an effective orthog-onal symmetric Lie algebra.

The next lemmas clarify the interplay between the previous definitions.

Lemma 2.1.11. Let (G,H) be a Riemannian symmetric pair with involutionσ. Assume either that G is semisimple or that (G,H) has been obtained withthe procedure described in Remark 2.1.5.Then the Lie algebra g of G becomes an effective orthogonal symmetric Liealgebra if we equip it with deσ.

Proof. The subalgebra Fix(deσ) = h is compactly embedded becauseAdG(H)is compact and it is precisely the identity component of the Lie subgroup ofInt(g) associated with adg(h).If G is semisimple, the centre of g is trivial and so must be its intersectionwith h. If G = I0(M) for some symmetric space M and H is the stabiliserof some point, it is clear that H cannot meet the centre of G outside e (bytransitivity of the action of G on M). This implies that h intersects triviallythe centre of g (thanks to Lemma 1.2.10).

Lemma 2.1.12. Let (g, s) be an effective orthogonal symmetric Lie algebrawith Fix(s) = h. Let G be a Lie group with Lie algebra g and H the connectedsubgroup associated with h.If G is simply connected, (G,H) is a Riemannian symmetric pair.

Proof. Since G is simply connected, Theorem 1.2.5 guarantees the existenceof an involution σ of G with deσ = s. Since H is connected, we musthave σ|H = id, as this holds in a neighbourhood of the identity (beingσ(expX) = exp(deσ(X))). In particular H is the identity component of theset of fixed points of σ and must therefore be closed.Moreover, AdG(H) is precisely the connected subgroup of Int(g) associatedwith adg(Fix(s)) and so has to be compact.

Lemma 2.1.13. Let (g, s) be an effective orthogonal symmetric Lie algebrawith Fix(s) = h. Let G be a Lie group with Lie algebra g and suppose thatthe connected subgroup H with Lie algebra h is closed.Then there exist G-invariant Riemannian metrics on G/H. Endowing G/Hwith one such metric, one gets a locally symmetric space.

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Proof. Since H is connected and h is compactly embedded, it is clear thatAdG(H) is compact. A G-invariant metric on G/H can then be constructedas in Remark 2.1.7 by fixing anAdG(H)-invariant scalar product on To(G/H).Now let π : G→ G be the universal covering space of G and call H the iden-tity component of π−1(H). It is clear that G/H → G/π−1(H) ' G/His a covering map; choosing an appropriate G-invariant metric on G/H,this map becomes locally isometric. Moreover, since G is simply connected,Lemma 2.1.12 guarantees that (G,H) is a Riemannian symmetric pair andRemark 2.1.7 implies that G/H becomes a symmetric space.

Actually, the space G/H is the universal covering of G/H: every loop inG/H can be lifted to an open path in G (since G→ G/H is a submersion)which can then be closed with a path in H; in particular every loop in G/His the image of a loop in G, which is simply connected.

Remark 2.1.14. The conclusion of Lemma 2.1.8 holds for G/H also in thesetting of Lemma 2.1.13, as is easy to check by looking at G/H.

Finally, we present the following fact without proof (see Theorem 4.1 inChapter V in [Hel78]). It explains the relationship between Remark 2.1.7 andRemark 2.1.5 when G is semisimple. Furthermore, we will need to appeal tothis result in the following section.

Theorem 2.1.15. Let (G,H) be a Riemannian symmetric pair. Assumethat G is semisimple and that H does not intersect the centre of G. Then, en-dowing the quotientM = G/H with a G-invariant metric as in Remark 2.1.7,the group G acts on M by isometries and we have:

G = I0(M) .

2.2 Decomposition of symmetric spaces

In Section 2.1 we have seen that a symmetric space M can always be rep-resented as a quotient G/H, where G is a connected Lie group and H is acompact subgroup. Moreover, G is endowed with an involution σ and deσturns g into an effective orthogonal symmetric Lie algebra with Fix(deσ) = h.

Definition 2.2.1. Let (g, s) be an effective orthogonal symmetric Lie alge-bra. If h and p are the eigenspaces of s relative to the eigenvalues 1 and −1respectively, we can write g = h⊕ p.

• If g is compact and semisimple, we say that (g, s) is of compact type;

• if g is semisimple, but not compact, we say that (g, s) is of noncompacttype if g = h⊕ p is a Cartan decomposition;

• if p is an abelian ideal, we say that (g, s) is of euclidean type.

33

The proof of the following fact is purely algebraic and can be found in[Hel78] as Theorem 1.1 of Chapter V.

Theorem 2.2.2. Let (g, s) be an effective orthogonal symmetric Lie algebraand let h and p be the eigenspaces of s relative to the eigenvalues 1 and −1.Let g be any adg(h)-invariant scalar product on p.Then there are ideals g0, g+, g− of g so that:

• g = g0 ⊕ g+ ⊕ g− and each piece is s-invariant;

• (g0, s|g0), (g+, s|g+) and (g−, s|g−) are effective orthogonal symmetricLie algebras, respectively of euclidean, compact and noncompact type.

Moreover, this decomposition is orthogonal with respect to both the Killingform of g and g.

Definition 2.2.3. Let M be a symmetric space. Then, setting G = I0(M)and denoting by Gp the subgroup of G fixing p ∈M , we have that (G,Gp) isa Riemannian symmetric pair and the Lie algebra g of G inherits a structureof orthogonal symmetric Lie algebra thanks to Lemma 2.1.11.According to the type of g, we say that M is of euclidean, compact or non-compact type. The same terminology applies to locally symmetric spaces,according to their universal covering space.

Corollary 2.2.4. A simply connected symmetric space M is always iso-metric to a product M0 ×M+ ×M−, where M0 is an euclidean space andM+,M− are simply connected symmetric spaces of compact and noncompacttypes, respectively.

Proof. We can write M = G/Gp with the notation of Definition 2.2.3 andconsider the associated orthogonal symmetric Lie algebra (g, s). Let π : G→G be the universal covering of G and denote by Gp the identity componentof π−1(Gp).It is clear that G/π−1(Gp) ' G/Gp, so the natural map G/Gp → G/Gp is acovering map; since M is simply connected, this map must be a diffeomor-phism (in particular π−1(Gp) is connected).The Lie algebra of Gp is h = Fix(s) and we denote by p the eigenspace ofs relative to the eigenvalue −1. The Riemannian metric on M correspondsto an adg(h)-invariant scalar product g on p. Theorem 2.2.2 then yields adecomposition g = g0 ⊕ g+ ⊕ g− associated with g; set hi = h ∩ gi andpi = p ∩ gi, for i ∈ 0,+,−.If G0, G+, G− are simply connected Lie groups with Lie algebras g0, g+, g−and H0, H+, H− are their connected subgroups associated with h0, h+, h−,then it is clear that G is isomorphic to G0 ×G+ ×G− and the isomorphismtakes Gp to H0 ×H+ ×H−, since h = h0 ⊕ h+ ⊕ h−. In particular

G/Gp ' G0/H0 ×G+/H+ ×G−/H−.

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Since the restriction of g to each pi is adg(hi)-invariant, it induces a Rie-mannian metric on Mi = Gi/Hi that turns it into a symmetric space; seeLemma 2.1.12 and Remark 2.1.7. Moreover, g(pi, pj) = 0 if i 6= j, so that weget an isometry between M and the Riemannian product M0 ×M+ ×M−.We have to check that each Mi is a symmetric space “of type i”. First, M0

is simply connected (since M is) and we shall see in the next section that itssectional curvature vanishes identically (Theorem 2.3.2); therefore, M0 is aneuclidean space.If i = ±, let Γi be the intersection of the centre of Gi with Hi; since Gi issemisimple, Γi is discrete. Observe that (Gi/Γi, Hi/Γi) is again a Rieman-nian symmetric pair inducing the symmetric spaceMi. Moreover, the centreof Gi/Γi is disjoint from Hi/Γi; indeed, if hΓi is in the centre of Gi/Γi, themap g 7→ ghg−1 on Gi takes values in Γi, which is discrete, and hence mustbe constant, i.e. h lies in Γi.Now we can invoke Theorem 2.1.15 to obtain I0(Mi) = Gi/Γi. In particularthe space Mi is of compact/noncompact type since the pair (Gi/Γi, Hi/Γi)is.

Of course, each of the factors M0, M+, M− can happen to be simply asingle point.Corollary 2.2.4 reduces the study of general symmetric spaces to those ofcompact and noncompact type; we will see in Section 2.4 that these corre-spond to compact and noncompact semisimple Lie groups, respectively. Inparticular, symmetric spaces of noncompact type are noncompact and thoseof compact type are compact.

2.3 Curvature of symmetric spaces

The proof of the following result is particularly technical and can be foundin [Hel78] (Theorem 4.2, Chapter IV).

Theorem 2.3.1. Let (G,H) be a Riemannian symmetric pair with involu-tion σ; assume that H is connected. Endow the quotient M = G/H witha G-invariant Riemannian metric (as in Remark 2.1.7) and define o ∈ Mas the projection of the identity element of G. Let g = h ⊕ p be the usualdecomposition of the Lie algebra of G into eigenspaces of deσ.Then, the curvature tensor at o is given by:

Ro(X,Y )Z = [[X,Y ], Z] ,

where X,Y, Z ∈ ToM can be seen as vectors in p ⊆ g.

Theorem 2.3.2. Let (G,H) be a Riemannian symmetric pair with involu-tion σ. Set s = deσ and assume that (g, s) is an efficient orthogonal symmet-ric Lie algebra; suppose moreover that H is connected. Endow Mn := G/H

35

with a G-invariant metric.Then, assuming that n ≥ 2:

1. if (g, s) is of compact type, M has nonnegative curvature;

2. if (g, s) is of noncompact type, M has nonpositive curvature;

3. if (g, s) is of euclidean type, M is flat, i.e. its universal covering spaceis isometric to Rn.

Proof. Since M is homogeneous, it suffices to compute the sectional curva-tures at o; here the curvature tensor is given by Theorem 2.3.1.When (g, s) is of euclidean type, Theorem 2.3.1 trivially implies that the sec-tional curvature of M vanishes everywhere; this is well-known to be equiva-lent to being covered by an euclidean space. So let us assume that we are inone of the first two cases; in particular, we can assume g to be semisimple.The metric g on ToM corresponds to an adg(h)-invariant metric on p (man-taining the notation of Definition 2.2.1). Hence there exists an endomor-phism f of p (as a vector space) satisfying

κ(X,Y ) = g(f(X), Y ), ∀X,Y ∈ p,

where κ is the Killing form of g. Since κ is symmetric, we have

g(f(X), Y ) = g(X, f(Y )), ∀X,Y ∈ p,

so that p is the direct sum of the eigenspaces pi of f and these spaces aremutually orthogonal with respect to g and κ.We have the easy relations [p, p] ⊆ h and [h, pi] ⊆ pi, for each i; the latteridentities follow from the fact that f commutes with all the elements ofadg(h), since g and κ are adg(h)-invariant.Observe that [pi, pj ] = 0 whenever i 6= j. Sure enough, take Xi ∈ pi, Xj ∈ pjand Y ∈ h and notice that κ([Xi, Xj ], Y ) = κ(Xj , [Y,Xi]) = 0; now [Xi, Xj ]has to be zero as the restriction of κ to h is negative definite (in both ourcases).To conclude, let λi be the eigenvalue of f associated with the eigenspace pi;it is immediate that λi < 0 in case 1 and λi > 0 in case 2, for each i. Nowtake X =

∑Xi and Y =

∑Yi in p, with Xi, Yi ∈ pi; we have

Ro(X,Y )X = [[X,Y ], X] = [∑i

[Xi, Yi], X] =∑i

[[Xi, Yi], Xi],

where the last equality is due to the fact that, if i 6= j:

[[Xi, Yi], Xj ] = −[[Yi, Xj ], Xi]− [[Xj , Xi], Yi] ∈ pj ∩ pi = 0.

36

Moreover, if V is generated by X,Y and they are orthonormal:

secV = g

(∑i

[[Xi, Yi], Xi] , Y

)=∑i

g ([[Xi, Yi], Xi], Yi) =

=∑i

1

λiκ ([[Xi, Yi], Xi], Yi) =

∑i

1

λiκ ([Xi, Yi], [Xi, Yi]) ,

and we conclude by recalling that κ is negative definite on h.

Corollary 2.3.3. A locally symmetric space of compact, noncompact andeuclidean type has, respectively, nonnegative, nonpositive and vanishing sec-tional curvature.

2.4 Maximal compact subgroups

Assume throughout this section that G is a connected semisimple Lie group.In the first part of this section we will clarify the interplay between thenotions of compactness for Lie algebras and Lie groups. Then we will turnto the study of maximal compact subgroups in semisimple Lie groups.

Lemma 2.4.1. Let g be a compact Lie algebra. Then g = gs ⊕ z, where z isthe centre of g and gs is compact and semisimple.

Proof. Let (·, ·) be a g-invariant scalar product, whose existence is guaran-teed by Lemma 1.3.4. Let gs be the subspace of g that is orthogonal to zwith respect to (·, ·). Since (·, ·) is g-invariant, it is clear that gs is an idealand that g = gs ⊕ z. Since gs has trivial centre, the argument used to proveLemma 1.3.6 shows that the Killing form of g is negative definite on gs. Sincegs is an ideal, its Killing form is negative definite as well. In particular, it iscompact and semisimple by Lemma 1.3.6.

Lemma 2.4.2. A Lie algebra g is compact if and only if it is the Lie algebraof a compact Lie group.

Proof. If g is the Lie algebra of a compact (connected) Lie group G, thenInt(g) = AdG(G) must be compact.Conversely, if g is compact, Lemma 2.4.1 guarantees that g = gs⊕z. Then gsis isomorphic to adgs(gs) = Der(gs) which is the Lie algebra of the compactLie group Int(gs), while, if k = dim z, the k-torus (S1)k has Lie algebraisomorphic to z. In particular g is the Lie algebra of the compact groupInt(gs)× (S1)k.

Theorem 2.4.3. The universal covering space of a compact, semisimple,connected Lie group is compact as well.

37

Proof. Let g be the Lie algebra of G and G and κ its Killing form. ByLemmas 2.4.2 and 1.3.6, −κ is positive definite, hence it can be extended toa left-invariant metric on G and G; this turns the covering map π : G → Ginto a local isometry. Since G is compact, the Riemannian metrics justdefined on G and G are complete.Note that also right translations in G (and G) are isometries. Indeed, since κis AdG(G)-invariant, each Cg is an isometry at e; actually, Cg is an isometryat every point:

〈X,Y 〉h = 〈dLh−1(X), dLh−1(Y )〉e = 〈dCgdLh−1(X), dCgdLh−1(Y )〉e =

=⟨dLgh−1g−1dCg(X), dLgh−1g−1dCg(Y )

⟩e

= 〈dCg(X), dCg(Y )〉Cg(h) .

Hence each Rg = LgCg−1 is an isometry.Let∇ be the Levi-Civita connection associated with our Riemannian metrics.If X,Y are left-invariant vector fields:

〈∇XX,Y 〉 =1

2

2X 〈X,Y 〉 − Y 〈X,X〉+

+ 〈[X,X], Y 〉 − 〈[X,Y ], X〉 − 〈[Y,X], X〉

= 0 .

In particular, ∇XX = 0 whenever X is a left-invariant vector field. Thismeans that the geodesics through the origin of G (and G) are precisely theone-parameter subgroups.Now suppose for the sake of contradiction that G is not compact. Since Gis a complete Riemannian manifold, it is an easy consequence of the Ascoli-Arzelà Theorem that there exists a ray γ through the origin e, i.e. a globallyminimising geodesic defined on [0,+∞), with γ(0) = e.The closure of the one-parameter subgroup π γ in G must be a compact,abelian Lie group; hence, it must be a torus thanks to Corollary 1.2.11. Inparticular, it is clear that there exists a sequence of real numbers tn → +∞with pn := πγ(tn)→ e. Since, for a sufficiently big n, the point pn lies in anevenly covered neighbourhood of e, we can find lifts en of e, with

d(γ(tn), en) = d(pn, e)→ 0,

where we denote simply by d the distances induced by the Riemannian met-rics on G and G.We will obtain a contradiction by proving that the image of γ must be con-tained in the centre of G, which is discrete since G is semisimple.Take any g ∈ G and define γg(t) := gγ(t)g−1, which must be another raythrough e; we want to prove that γ and γg coincide. Since en lies in thecentre of G by Lemma 1.1.2, we have

d(γ(tn), γg(tn)) ≤ d(γ(tn), en) + d(geng−1, γg(tn))→ 0.

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If the velocities of γ, γg at e were distinct, we would be able to find ε > 0with ε < 2− d(γ(−1), γg(1)). Then, choosing n so that d(γ(tn), γg(tn)) < ε,we would have

tn + 1 = d(γ(−1), γ(tn)) ≤

≤ d(γ(−1), γg(1)) + d(γg(1), γg(tn)) + d(γg(tn), γ(tn)) <

< 2 + d(γg(1), γg(tn)) = 2 + (tn − 1) = tn + 1,

which is the sought contradiction.

Corollary 2.4.4. If g is semisimple and compact, all Lie groups having Liealgebra g are compact.

Proof. Lemma 2.4.2 and Theorem 2.4.3 imply that the simply connected Liegroup with Lie algebra g is compact. The statement follows at once.

Remark 2.4.5. In particular, notice that, if (G,H) is a Riemannian sym-metric pair of compact type, G must be a compact Lie group. So all sym-metric spaces of compact type are compact.On the other hand, if (G,H) has noncompact type, G cannot be compact andneither will be G/H (assuming H compact). Hence, all symmetric spaces ofnoncompact type are noncompact.

The following result is an improvement of Lemmas 2.1.12 and 2.1.13 foralgebras of noncompact type.

Proposition 2.4.6. Let (g, s) be an effective orthogonal symmetric Lie al-gebra of noncompact type, G any connected Lie group with Lie algebra g andH a subgroup with Lie algebra h = Fix(s). Then:

1. H is closed, connected and contains the centre of G;

2. (G,H) is a Riemannian symmetric pair.

Proof. Let H0 be the identity component of H and let g = h⊕p be a Cartandecomposition.We start by noticing that H0 is closed. Indeed, since h is compactly embed-ded in g, the groupAdG(H0) must be compact; in particular, Ad−1

G (AdG(H0))is closed in G and its identity component is precisely H0, since the centre ofG is discrete.Now observe that the map

p×H0 → G

(X,h) 7→ (expX)h

is surjective. Indeed, the map

p→ G/H0

X 7→ π(expX)

39

is surjective thanks to Lemma 2.1.13 and Remark 2.1.14.However, the map

p×H → G

(X,h) 7→ (expX)h

is injective. Indeed, if we have (expX1)h1 = (expX2)h2, we also get, bytaking AdG of both sides:

exp(adgX1)AdG(h1) = exp(adgX2)AdG(h2).

Definingκs(X,Y ) := −κ(X, s(Y )),

where κ is the Killing form of g, we obtain a scalar product. In a κs-orthonormal basis, each AdG(hi) is represented by an orthogonal matrix,each adgXi by a symmetric one and each exp(adgXi) by a positive-definite,symmetric one. In particular, we must have

exp(adgX1) = exp(adgX2),

AdG(h1) = AdG(h2),

which implies adgX1 = adgX2 and X1 = X2 since g has a trivial centre;so expX1 = expX2 and from (expX1)h1 = (expX2)h2 we get h1 = h2,completing the proof of injectivity.In particular, the injectivity of p×H → G and the surjectivity of p×H0 → Gimply that H = H0, i.e. H is connected; in particular, H is closed.Exchanging H with Ad−1

G (AdG(H)) and repeating the previous arguments,we see that Ad−1

G (AdG(H)) must be connected. Hence H = Ad−1G (AdG(H)),

which means that H has to contain the centre of G.Finally, let G be the universal covering group of G and H its connectedsubgroup with Lie algebra h. Lemma 2.1.12 shows that (G, H) is a symmetricRiemannian pair; call σ the corresponding involution of G and H ′ its fixed-point set. Thanks to the previous discussion, H ′ must contain the centreof G (and coincides with H); in particular there exists a discrete centralsubgroup Γ / H ′, so that G = G/Γ and σ descends to an involution on Gthat turns (G,H) into a Riemannian symmetric pair.

Theorem 2.4.7. Let (G,H) be any Riemannian symmetric pair of noncom-pact type, with G connected; let g = h⊕ p be a Cartan decomposition with hthe Lie algebra of H. Then the map

Φ: p×H → G

(X,h) 7→ (expX)h

is a diffeomorphism. In particular, the space G/H is diffeomorphic to aeuclidean space.

40

The proof of the previous fact is a little involved and we omit it; see forexample [Hel78], Chapter VI, Theorem 1.1. Nevertheless, we remark thatthe fact that the above map is a smooth bijection follows from the first partof the proof of Proposition 2.4.6 and from the general theory of differentialequations.The following proposition shows that a semisimple Lie group is always asso-ciated to a symmetric space.

Proposition 2.4.8. Given G semisimple and connected, with Lie algebra g,there are only two possibilities:

1. G is compact;

2. g has a Cartan decomposition g = h⊕ p with h 6= g and defining

s : h⊕ p→ h⊕ p

T +X 7→ T −X

we obtain an order-two automorphism of g such that (g, s) is an effec-tive orthogonal symmetric Lie algebra of noncompact type. Moreover,G has maximal compact subgroups and they are all conjugated.

Proof. If g is compact, we are in the first case thanks to Corollary 2.4.4.Otherwise, given a Cartan decomposition g = h⊕ p, Lemma 1.3.13 guaran-tees that h is compactly embedded; moreover, since [h, p] ⊆ p and [p, p] ⊆ h,it is easy to check that s is an involutive automorphism. Hence (g, s) is anorthogonal symmetric Lie algebra of noncompact type.Let H be a subgroup of G with Lie algebra h; Proposition 2.4.6 guaranteesthat H is closed and connected. Since h is compact due to Lemma 1.3.5, wecan invoke Lemma 2.4.1 and split it as h = hs⊕ z, where z is the centre of h.Let Hs, Z be the connected Lie subgroups with Lie algebras hs and z re-spectively; they are both closed in H (and hence in G) since Hs is com-pact (Corollary 2.4.4) and Z is the identity component of the centre of H(Lemma 1.2.10). Thanks to Corollary 1.2.11, Z must be the product of atorus T and an euclidean space V , both closed subgroups of G.The set K = HsT is a compact subgroup of G. Note that K ∩ V = e,as V does not have any nontrivial compact subgroups; since V commuteselement-wise with K and H is connected, we must have H = K × V . Inparticular, K is a maximal compact subgroup of H.If K ′ is any other compact subgroup of G, it acts by left-translations onG and by isometries on G/H. Endowing G/H with a G-invariant metric,we obtain a contractible Riemannian manifold with nonpositive curvature(Theorems 2.4.7 and 2.3.2) and Theorem 1.4.7 implies the existence of afixed point xH for the action of K ′. This can be rephrased as x−1K ′x ⊆ H,or, more precisely, x−1K ′x ⊆ K. In particular, all compact subgroups of Gare conjugated to a subgroup of K.

41

We now see that K is a maximal compact subgroup also in G. If therewere a compact subgroup of K ′ < G properly containing K, we would havex−1K ′x ⊆ K for some x ∈ G. In particular, K and K ′ would have the samedimension and K ′ would have to be disconnected, its identity componentbeing K; this is clearly absurd.The previous discussion also shows the uniqueness of compact maximal sub-groups up to conjugation.

Corollary 2.4.9. Let G be a semisimple connected Lie group and K a max-imal compact subgroup (whose existence is guaranteed by Proposition 2.4.8).Then G/K is contractible.

Proof. If we are in the first case of Proposition 2.4.8 the result is obvious.In the second case, we can assume, with the notation of Proposition 2.4.8,that K = HsT and H = K × V . Then, by Theorem 2.4.7,

G/K ' p×H/K ' p× V.

42

Chapter 3

Continuous Group Cohomology

In this chapter we will be concerned with continuous group cohomology oftopological groups. We will define it for fairly general groups in Section 3.1and show that it can be computed exploiting particular resolutions of Fréchetmodules; an almost identical treatment can be given also in the context ofusual group cohomology, but we will not discuss it. Then, we will brieflyconsider a few matters regarding functoriality (Section 3.2) and the case ofdiscrete groups (Section 3.3).In Section 3.4 our focus will shift to Lie groups. For a group G we willshow that its cohomology can be computed via complexes which are relatedto the geometry of its coset spaces. The main result of this kind will beVan Est’s Theorem which will prove useful in Chapters 5 and 6. As anotherconsequence, we will see that, under suitable hypotheses, the cohomology ofG is finite-dimensional and vanishes in high dimension.

3.1 Continuous Group Cohomology

We assume throughout this section, that G is a locally compact, Hausdorff,second-countable topological group. We will define the continuous cohomol-ogy spaces only in this context.

Definition 3.1.1. Let E be a Fréchet G-module. A Fréchet resolution(E,E∗, d∗) of E is an exact complex

0 // Eε // E0 d0 // E1 d1 // E2 d2 //// ...

where each Ek is a Fréchet G-module and all maps are morphisms of G-modules. A Fréchet resolution is said to be strong if all the maps involvedare strong (in the sense of Definition 3.1.1).

Note in particular that the notion of strength of a resolution does notdepend on the action of G on the involved spaces.

43

Definition 3.1.2. Given a Fréchet resolution (E,E∗, d∗), a contracting ho-motopy s∗ is a set of linear continuous maps skk≥0 (not necessarily G-equivariant):

0 // Eε // E0

s0

d0 // E1 d1 //

s1yy

E2 d2 ////

s2yy

...

s3yy

satisfying dk−1sk + sk+1dk = id for each k ≥ 0, where the morphism d−1 hasto be understood as ε.

Proposition 3.1.3. A Fréchet resolution (E,E∗, d∗) of a G-module E isstrong if and only if it admits a contracting homotopy.

Proof. First, assume the existence of a contracting homotopy s∗. Call πn theprojection from En to En/Ker dn and ιn the inclusion Ker dn → En. Thena left inverse for ιn is provided by dn−1sn, while πnsn+1 is a left inverse fordn : En/Ker dn → En+1 and s0 is a left inverse for ε.Conversely, assume that (E,E∗, d∗) is strong and choose continuous linearmaps αn, with αnιn = id. We have splittings En = Ker dn ⊕ Ker αnand dn−1 gives a homeomorphism between Ker αn−1 and Ker dn, thanksto Lemma 1.5.7. Hence we can define sn|Ker dn = (dn−1|Ker αn−1)−1 andsn|Ker αn = 0. It is easy to check that this is the required homotopy.

Definition 3.1.4. A Fréchet G-module E is relatively injective if, wheneverA,B are Fréchet G-modules and u : A → E, v : A → B are G-maps, with vstrong and injective, there exists a G-map w such that the following diagramis commutative:

A B

E .

v

u w

A Fréchet resolution (E,E∗, d∗) is relatively injective if each En is, for n ≥ 0.

The previous definition of relative injectivity is valid in the categoryof Fréchet G-modules, which will be the only category which we are con-cerned with in this chapter. In Chapter 4 however, we will meet a differentnotion of relative injectivity, in the category of Banach G-modules (see Def-inition 4.1.1).The definitions of relative injectivity and strength of a resolution are devisedto yield the following result, which will be our main tool in the definition ofcontinuous cohomology.

Proposition 3.1.5. Let f : E → F be a G-map between Fréchet G-modulesand let (E,E∗, d∗), (F, F ∗, δ∗) be Fréchet resolutions of E and F .If (E,E∗, d∗) is strong and (F, F ∗, δ∗) is relatively injective, then f can beextended to a chain map made of G-morphisms. This extension is unique upto homotopies made of G-morphisms.

44

Proof. First, we have to construct G-morphisms fn : En → Fn for n ≥ 0 sothat the diagram

0 // Eε //

f

E0 d0 //

f0

...dn−1// En

dn //

fn

En+1 dn+1//

fn+1

...

0 // Fη // F 0 δ0 // ...

δn−1// Fn

δn // Fn+1 δn+1// ...

commutes; we will prove this by induction on n.Since ε is strong and injective and F 0 is relatively injective, there existsf0 satisfying f0ε = ηf . Now, suppose that we have contructed the mapsf0, ..., fn and that δkfk = fk+1dk for k = 0, ..., n − 1. The map δnfn

vanishes on Ker dn = Im dn−1 since δnfndn−1 = δnδn−1fn−1 = 0 (if n = 0,we have to interpret d−1 and δ−1 as ε and η, respectively); hence we obtain awell-defined map δnfn : En/Ker dn → Fn+1. Since dn : En/Ker dn → En+1

is strong and injective and Fn+1 is relatively injective, we can find a G-mapfn+1 : En+1 → Fn+1 with fn+1dn+1 = δnfn. The same map then satisfiesfn+1dn+1 = δnfn.Suppose now that gnn∈N is another family of G-maps extending f ; weshall construct G-maps T 0 : E0 → F and Tn : En → Fn−1 for n ≥ 1, so thatgn − fn = Tn+1dn + δn−1Tn, for each n ≥ 0. Without loss of generality, wecan assume that f ≡ 0 and fn ≡ 0 by simply replacing each gn with gn−fn.

0 // Eε //

f

E0 d0 //

T 0

~~f0

...dn−1

// Endn //

Tn

~~

fn

En+1 dn+1

//

Tn+1

||fn+1

...

0 // Fη // F 0 δ0 // ...

δn−1// Fn

δn // Fn+1 δn+1

// ...

Again we will proceed by induction on n; start by simply choosing T 0 ≡ 0.Assume that we have constructed T 0, ..., Tn so that gk = T k+1dk + δk−1T k

for k ≤ n− 1. The map gn− δn−1Tn vanishes on Ker dn = Im dn−1; indeed:

(gn − δn−1Tn)dn−1 = δn−1gn−1 − δn−1(gn−1 − δn−2Tn−1) = 0,

where δ−2 and g−1 have to be understood as zero maps. Then, the same argu-ment used above guarantees the existence of some G-map Tn+1 : En+1 → Fn

satisfying gn − δn−1Tn = Tn+1dn.

Let now E be a Fréchet G-module. We will later prove that a strong,relatively injective resolution of E always exists. If (E,E∗, d∗) is one suchresolution, we can consider the complex ((E∗)G, d∗) obtained by restrictingeach coboundary map to the invariant subspaces (and removing E):

0 // (E0)Gd0 // (E1)G

d1 // ...dn−1// (En)G

dn // ... .

45

Definition 3.1.6. Given a complex of Fréchet spaces

0 // E0 d0 // E1 d1 // ...dn−1// En

dn // ...

with linear, continuous coboundary maps, we define its cohomology spaces

Hk(E∗) := (Ker dk)/

(Im dk−1).

These are vector spaces endowed with a topology (the quotient topology),but will not always be topological vector spaces; sure enough, the topologyon Hk(E∗) is Hausdorff if and only if Im dk−1 is closed.

Corollary 3.1.7. Let E be a Fréchet G-module and (E,E∗, d∗), (E,F ∗, δ∗)two strong, relatively injective resolutions.Then the cohomology spaces of the subcomplexes ((E∗)G, d∗) and ((F ∗)G, δ∗)are isomorphic as vector spaces equipped with a topology.

Proof. Thanks to Proposition 3.1.5 we can find, for each n ≥ 0, G-mapsfn : En → Fn and gn : Fn → En extending id : E → E and commuting withthe coboundary maps. The same proposition also guarantees the existenceof G-maps Tn : En → En−1, with gnfn = Tn+1dn + dn−1Tn.By restriction, all these maps define maps between the subspaces of invari-ants of the involved G-modules. In particular we get fn : (En)G → (Fn)G

and gn : (Fn)G → (En)G, with T ∗ defining a homotopy between g∗f∗ andthe identity of E∗; similarly we obtain a homotopy between f∗g∗ and theidentity of F ∗.Hence the maps that f∗ and g∗ induce in cohomology are linear homeomor-phisms.

The previous result finally allows us to define continuous cohomology.

Definition 3.1.8. The k-th continuous cohomology space of G with coef-ficients in the Fréchet G-module E is the k-th cohomology space of thecomplex ((E∗)G, d∗), where (E,E∗, d∗) is any strong, relatively injective res-olution of E.This space will be denoted by Hk

c (G,E) and is endowed with a well-definedtopology.

In general, the spaces Hkc (G,E) will not be Fréchet spaces (nor topo-

logical vector spaces), since the topology that they inherit could not beHausdorff. Nonetheless, we have the following result.

Lemma 3.1.9. If Hkc (G,E) is finite-dimensional, its topology is Hausdorff.

Proof. Let (E,E∗, d∗) be a strong, relatively injective resolution of E. IfZk := (Ker dk) ∩ (Ek)G, the map

dk−1 : (Ek−1)G → Zk

has closed image because of Corollary 1.5.10. It follows from Lemma 1.5.5that Hk

c (G,E) is a Fréchet space with the induced topology.

46

Remark 3.1.10. Proposition 3.1.5 also implies that aG-morphism f : E → Fbetween Fréchet G-modules always induces a continuous map

f∗ : H∗c (G,E)→ H∗c (G,F )

in cohomology. Moreover, the association f → f∗ is clearly functorial andcovariant.

We shall now prove the existence of a strong, relatively injective resolu-tion for every Fréchet G-module E. The resolution that we will construct isusually called the standard resolution of E.Consider the complex

0 E C(G,E) ... C(Gn+1, E) ... ,ε d0 dn−1 dn

where the coboundary maps are

dnf(g0, ..., gn+1) :=

n+1∑j=0

(−1)jf(g0, ..., gj , ..., gn)

and each C(Gk+1, E) is a Fréchet G-module as in Example 1.5.15, with Gacting on Gk+1 by:

g · (g0, ..., gk) := (gg0, ..., ggk).

It is immediate to check that d2 = 0 and that all the involved maps aremorphisms of G-modules.

Proposition 3.1.11. The complex above is a strong resolution of E.

Proof. It suffices to prove the existence of a contracting homotopy s∗, sincethis also implies exactness. Define for f ∈ C(Gn+1, E):

(snf)(g0, ..., gn−1) := f(e, g0, ..., gn−1),

where e is the identity element of G, and s0f = f(e) ∈ E. These maps areplainly linear and continuous; moreover,

(dn−1snf)(g0, ..., gn) =n∑j=0

(−1)j · (snf)(g0, ..., gj , ..., gn) =

=

n∑j=0

(−1)j · f(1, g0, ..., gj , ..., gn) = f(g0, ..., gn)− (dnf)(1, g0, ..., gn) =

= (f − sn+1dnf)(g0, ..., gn).

47

Proposition 3.1.12. The Fréchet G-module C(Gk+1, E) is relatively injec-tive for each k ≥ 0.

Proof. Let A,B, u, v be as in Definition 3.1.4 and let s be a left inverse forthe map v. Define, for b ∈ B:

w(b)(g0, ..., gk) := u(g0 · s(g−1

0 · b))

(g0, ..., gk) .

The map w is well defined because w(b) is a continuous function of g0, ..., gk;indeed, if (g0,n, ..., gk,n) → (g0, ..., gk), we have that u(g0,n · s(g−1

0,n · b)) con-verges to u(g0·s(g0

−1·b)), uniformly on a compact neighbourhood of (g0, ..., gk).Hence, w(b)(g0,n, ..., gk,n) must converge to w(b)(g0, ..., gk).Moreover, w is itself continuous. Otherwise, there would be vectors bn → bsuch that w(bn) does not converge uniformly to w(b) on some compact setK ⊆ Gk+1. So we could find (g0,n, ..., gk,n)→ (g0, ..., gk) in K, with

w(bn)(g0,n, ..., gk,n) 6→ w(b)(g0, ..., gk);

this is absurd since u(g0,n · s(g−10,n · bn)) tends to u(g0 · s(g0

−1 · b)) uniformlyon every compact neighbourhood of (g0, ..., gk).Clearly w v = u, since

w (v(a)) (g0, ..., gk) = u(g0 · s(g−1

0 · v(a)))

(g0, ..., gk) =

= u(g0 · sv(g−1

0 · a))

(g0, ..., gk) = u(a)(g0, ..., gk).

Finally, we check that w is G-equivariant:

(g · w(b)) (g0, ..., gk) = g ·[w(b)(g−1g0, ..., g

−1gk)]

=

= g ·[u(g−1g0 · s(g−1

0 g · b))(g−1g0, ..., g−1gk)

]=

= u(gg−1g0 · s(g−1

0 g · b))

(g0, ..., gk) =

= w(g · b)(g0, ..., gk).

Corollary 3.1.13. The continuous cohomology H∗c (G,E) can be computedas the cohomology of the complex:

0 C(G,E)G C(G2, E)G ... C(Gn+1, E)G ... .d0 d1 dn dn

Hu’s original definition of continuous group cohomology was via the com-plex that appears in the previous corollary ([Hu54]). However, the categoricalapproach to cohomology that we adopted in the present section is mainlydue to Hochschild and Mostow ([HM62]).

48

3.2 Functoriality

We have already seen in Remark 3.1.10 that the association E 7→ H∗c (G,E)is a covariant functor. In this section we will be concerned with the functo-riality of G 7→ H∗c (G,E).Suppose that H,G are locally compact, Hausdorff, second-countable topo-logical groups and let f : H → G be a continuous homomorphism. EveryFréchet G-module has an induced structure of H-module given by:

h · e := f(h) · e ;

we will denote by f∗E (or simply by E) the induced H-module.We can define a continuous chain map (C(G∗+1, E), d∗)→ (C(H∗+1, f∗E), d∗)extending id : E → f∗E; simply consider the maps

fk : C(Gk+1, E)→ C(Hk+1, f∗E) ,

given by the composition with

fk : Hk+1 → Gk+1

(h0, ..., hk) 7→ (f(h0), ..., f(hk)) .

Moreover, viewing the C(Gk+1, E) as theH-modules f∗C(Gk+1, E), the mapsfk are H-equivariant.Hence we obtain maps in cohomology f∗ : Hk

c (G,E)→ Hkc (H, f∗E), induced

by the composition

C(Gk+1, E)G → C(Gk+1, E)H → C(Hk+1, f∗E)H .

If g : G→ K is another continuous homomorphism, with K locally compact,Hausdorff and second-countable, we clearly have (g f)∗ = f∗ g∗ and(idH)∗ = id.

Remark 3.2.1. Let now (E,E∗, δ∗) and (f∗E,F ∗, δ∗) be two more strong,relatively injective resolutions of E and f∗E, respectively. Suppose that weare given a continuous chain map

φ∗ : (E∗, δ∗)→ (F ∗, δ∗)

that extends id : E → f∗E and is H-equivariant when we view each Ek asthe H-module f∗Ek.Then we have another map in cohomology φ∗ : Hk

c (G,E) → Hkc (H, f∗E),

induced by the composition

(Ek)G → (Ek)H → (F k)H .

We will soon show that the maps in cohomology f∗ and φ∗ coincide.

49

In the setting of the previous remark, observe that we can constructanother H-equivariant, continuous chain map (E∗, δ∗) → (F ∗, δ∗). Simplycompose f∗ : (C(G∗+1, E), d∗) → (C(H∗+1, f∗E), d∗) on the right with a G-equivariant chain map (E∗, δ∗) → (C(G∗+1, E), d∗) extending idE and onthe left with an H-equivariant chain map (C(H∗+1, f∗E), d∗) → (F ∗, δ∗)extending idF (the existence of which is guaranteed by Proposition 3.1.5).

Lemma 3.2.2. Let (E,E∗, δ∗) and (f∗E,F ∗, δ∗) be strong, relatively injec-tive resolutions of E and f∗E, respectively. Suppose we have H-equivariant,continuous chain maps φ∗, ψ∗ : (E∗, δ∗)→ (F ∗, δ∗) extending id : E → f∗E.Then φ∗, ψ∗ induce the same map H∗c (G,E)→ H∗c (H, f∗E) in cohomology.

Proof. Viewing each Ek as the H-module f∗Ek, we obtain a strong (not nec-essarily relatively injective) resolution of f∗E. Proposition 3.1.5 then pro-vides an H-invariant homotopy between the chain maps φ∗ and ψ∗. Hence,φ∗, ψ∗ induce the same map between the cohomologies of the complexes((E∗)H , δ∗) and ((F ∗)H , δ∗).Since the maps H∗c (G,E) → H∗c (H, f∗E) are obtained by composing thesemaps with the maps H∗c (G,E) → H∗((E∗)H , δ∗) induced by the inclusion(E∗)G → (E∗)H , the assertion follows.

From the previous lemma and Remark 3.2.1 we obtain the followingresult.

Corollary 3.2.3. In the setting of the previous lemma, the map induced byφ∗ and ψ∗ in cohomology is precisely

f∗ : Hkc (G,E)→ Hk

c (H, f∗E).

3.3 Discrete groups

In this section we shall briefly review what simplifications can be made inthe context of discrete groups. Throughout this section Γ will be a countablegroup endowed with the discrete topology.First, observe that a Fréchet Γ-module is simply a Fréchet space E equippedwith a left action of Γ by homeomorphisms; all the additional continuityrequirements are automatically satisfied.The standard resolution simply becomes

0 R C0(Γ, E) C1(Γ, E) C2(Γ, E) ... ,ε d0 d1 d2

whereCk(Γ, E) := f : Γk+1 → E

50

is endowed with the usual action.Hence the continuous cohomology H∗c (Γ, E) is computed simply by the com-plex

0 C0(Γ, E)Γ C1(Γ, E)Γ C2(Γ, E)Γ ... .d0 d1 d2

So, forgetting about the topology on H∗c (Γ, E), the continuous cohomologyof Γ is isomorphic to the usual group cohomology H∗(Γ, E).The definition of the latter does not require any topology on the vector spaceE, but simply a left action of G by linear isomorphisms. See for example[Bro82] for more details on group cohomology.

3.4 Van Est’s Theorem

Throughout this section G will be a connected Lie group.In this case, given a Fréchet G-module E, we can find an analogue of thestandard resolution of E involving E-valued smooth maps instead of contin-uous one. This will also allow us to construct a resolution of E involvingdifferential forms on a suitable manifold; this resolution will prove particu-larly useful since, as we shall see, the associated complex of invariants willbe degenerate, i.e. with vanishing coboundary maps.To avoid technicalities regarding the smoothness of E-valued functions, wewill carry out the corresponding proofs only in the case E = R with G actingtrivially. We remark nevertheless that all the results in the present sectioncan be adapted to Fréchet coefficients with no substantial changes (especiallyif E is finite-dimensional).

Proposition 3.4.1. Let G′ be a Lie subgroup of G. Then, C∞(G,R) is arelatively injective Fréchet G′-module, with G′ acting on G by left transla-tions.

Proof. We consider the space C∞(G,R) endowed with the structure of FréchetG′-module given in Remark 1.5.16.Let A,B, u, v be as in Definition 3.1.4 and let s be a left inverse for v. Fixmoreover a left Haar measure µ on G and a smooth, compactly supportedfunction ρ : G→ [0,+∞) with

∫ρ dµ = 1.

Definew(b)(x) :=

∫Gρ(x−1g)

[u(g · s(g−1 · b))(x)

]dµ(g) .

Since the integrand is a continuous function of g and, for each x, we areintegrating it only on a compact subset of G, it is clear that the value w(b)(x)is always well-defined and the function w(b) is plainly smooth in x.Moreover, w is continuous since all partial derivatives of w(b) are obtainedby integrating partial derivatives of ρ and u(g · s(g−1 · b)) and each of these

51

is continuous with respect to b in the topology of uniform convergence oncompact sets.Observe that w v = u, as

w (v(a)) (x) =

∫Gρ(x−1g)

[u(g · s(g−1 · v(a))

)(x)]dµ(g) =

=

∫Gρ(x−1g)

[u(g · sv(g−1a)

)(x)]dµ(g) =

∫Gρ(x−1g) [u(a)(x)] dµ(g) =

=

∫Gρ(g) [u(a)(x)] dµ(g) = u(a)(g0, ..., gk).

To conclude our proof, for h ∈ G′:

(h · w(b)) (x) = h ·[w(b)(h−1x)

]=

=

∫Gρ(x−1hg)

[u(g · s(g−1 · b)

)(h−1x)

]dµ(g) =

=

∫Gρ(x−1g)

[u(h−1g · s(g−1h · b)

)(h−1x)

]dµ(g) =

=

∫Gρ(x−1g)

[u(g · s(g−1h · b)

)(x)]dµ(g) = w(h · b)(x),

where we have repeatedly used that G acts trivially on R and that µ isleft-invariant.

Corollary 3.4.2. Whenever G′ is a Lie subgroup of G, the resolution ofFréchet G′-modules

0 R C∞(G,R) ... C∞(Gn+1,R) ... ,ε d0 dn−1 dn

with the same coboundary maps as in the standard resolution, is a strong,relatively injective resolution of the trivial G′-module R.

Proof. It is evident that the coboundary maps are continuous also in thiscontext. Moreover, the proof that we gave of Proposition 3.1.11 carries overto this case, since the same contracting homotopy is continuous also betweenthese spaces.Finally, G′ acts on Gk+1 by left translation in each factor; this is preciselythe action by left multiplication of the diagonal subgroup

(x, x, ..., x) ∈ Gk+1 | x ∈ G′ .

So Proposition 3.4.1 guarantees that theG′-module C∞(Gk+1,R) is relativelyinjective for each k ≥ 0.

52

Recall that Example 1.5.17 gives a structure of Fréchet G-module tothe spaces Ωp(G,R) of real differential forms, where G acts on G by lefttranslations.

Proposition 3.4.3. If G′ is a Lie subgroup of G, the G′-modules Ωp(G,R),p ≥ 0 are relatively injective.

Proof. First, notice that Ωp(G,R) is isomorphic, as a Fréchet G′-module, toC∞(G,R)⊗ΛpT ∗eG, where G′ acts trivially on ΛpT ∗eG and the tensor productmust be understood as in Example 1.5.18.Indeed, choose a basis ω1, ..., ωN of ΛpT ∗eG and construct G-invariant smoothp-forms ω1, ..., ωN on G by setting

(ωi)g [X1, ..., Xp] := ωi[(dLg−1)g(X1), ..., (dLg−1)g(Xp)

]at each g ∈ G. Since (ω1)q, ..., (ωp)q are a basis of ΛpT ∗qG for each q ∈ G,whenever ω is a p-form on G, there must exist functions λ1, ..., λN satisfying

ωq = λ1(q)(ω1)q + ...+ λN (q)(ωN )q .

It is clear that λi belongs to C∞(G,R) for each i, and this proves our claimsince:

g · ω = (τ(g−1))∗ω =N∑i=1

(λi τ(g−1)

)· (τ(g−1))∗ωi =

N∑i=1

(g · λi) · ωi .

In particular Ωp(G,R) is isomorphic, as a G′-module, to the direct sum ofa finite number of copies of C∞(G,R); since the latter is relatively injective,so must be the former.

For example, if G is a contractible smooth manifold (Nil and Sol arenontrivial examples, see [Sco83]), we can compute its continuous cohomologyvia the complex of G-invariant smooth real forms, as the next result implies.

Lemma 3.4.4. Let N be a contractible smooth manifold. Then the De Rhamcomplex

0 R C∞(N,R) Ω1(N,R) Ω2(N,R) ...ε d0 d1 d2

is a strong resolution of R made of Fréchet spaces.

Proof. All the spaces involved are Fréchet and the differentials are continuousthanks to Example 1.5.17. We will construct a contracting homotopy s∗.Let F : N × [0, 1]→ N a smooth map with F0 = idN and F1 collapsing thewhole manifold on some point x0. Moreover, let

Ip : Ωp(N × [0, 1],R)→ Ωp−1(N,R)

53

be the map given by integrating forms along the fibres ∗ × [0, 1].It is well-known that, setting sk = Ik F ∗, the relations id = sk+1dk+dk−1sk

hold for each k ≥ 0; see for example Lemma 17.9 in [Lee12]. Furthermore,it is easy to check that each sk is continuous.

In the remaining part of this section, we will always assume that H is acompact subgroup of G and we will denote byM the smooth manifold G/H.

Proposition 3.4.5. Let π be the projection π : G→M and π(k) : Gk →Mk

the map that is π on each factor. Then:

1. for each k ≥ 0, the map (π(k+1))∗ : C∞(Mk+1,R) → C∞(Gk+1,R) hasa left inverse that is a G-map;

2. for each p ≥ 0, the map π∗ : Ωp(M,R) → Ωp(G,R) has a left inversethat is a G-map.

Proof. 1. We can assume that k = 0 since the general case then followsby replacing G and H with Gk and Hk respectively.The map π∗ induces an isomorphism of Fréchet G-modules betweenC∞(M,R) and the following closed subspace of C∞(G,R):

C∞H (G,R) := f ∈ C∞(G,R) | f(xh) = f(x), ∀x ∈ G, ∀h ∈ H .

Therefore we simply have to define a G-equivariant retraction r ofC∞(G,R) onto C∞H (G,R). Let ν be the Haar measure on H withν(H) = 1. Given f : G→ R, simply define

(rf)(x) =

∫Hf(xh) dν(h) ,

which clearly is a smooth function on G and is constant on the rightcosets of H. The map r is continuous and rf = f whenever f lies inC∞H (G,R). Finally:

r(g · f)(x) =

∫H

(g · f)(xh) dν(h) =

∫Hf(g−1xh) dν(h) =

= (rf)(g−1x) = (g · (rf))(x).

2. Again π∗ allows us to identify isomorphically Ωp(M,R) with

ΩpH(G,R) := ω ∈ Ωp(G,R) | R∗hω = ω, ∀h ∈ H.

Sure enough, π∗ maps Ωp(M,R) into ΩpH(G,R) and, if ω ∈ Ωp

H(G,R),we can define a p-form π∗ω on M in the following way. Fix a left-invariant Riemannian metric on G which is also right-H-invariant (thisis possible by an averaging procedure since H is compact). Given

54

X1, ..., Xp ∈ TgHM and g′ ∈ gH, there are unique lifts X(g′)1 , ..., X

(g′)p

to Tg′G which are orthogonal to the manifold gH; moreover, since Hacts isometrically by right-translations on G, each dRh must send eachXg′

i to Xg′hi . In particular, we can define

(π∗ω)gH[X1, ..., Xp] := ωg′

[X

(g′)1 , ..., X(g′)

p

]and the result does not depend on which g′ we have chosen; also, π∗ω isa smooth p-form on M . It is easy to check that π∗ provides an inversefor π∗ and that both maps are continuous and respect the action of G.Now we have to construct a G-equivariant retraction r of Ωp(G,R)onto Ωp

H(G,R). Given a p-form ω on G, set:

(rω)g [X1, ..., Xp] :=

∫H

(R∗hω)g [X1, ..., Xp] dν(h).

Observe that:

(R∗h′(rω))g [X1, ..., Xp] = (rω)gh′ [dRh′(X1), ..., dRh′(Xp)] =

=

∫H

(R∗hω)gh′ [dRh′X1, ..., dRh′Xp] dν(h) =

=

∫H

(R∗h′R∗hω)g [X1, ..., Xp] dν(h) =

∫H

(R∗h′hω)g [X1, ..., Xp] dν(h) =

=

∫H

(R∗hω)g [X1, ..., Xp] dν(h) = (rω)g [X1, ..., Xp] .

All the other verifications are straightforward.

Lemma 3.4.6. Let E be a closed subspace of a Fréchet G-module F . As-sume that F is relatively injective and that there exists a G-map r : F → Esuch that r|E = id. Then E is relatively injective.In particular, C∞(Mk+1,R) and Ωp(M,R) are relatively injective G′-modules,for each Lie subgroup G′ of G.

Proof. As usual, let A,B be Fréchet G-modules and u : A→ E and v : A→B morphisms of G-modules, with v strong and injective. Call ι : E → Fthe inclusion map. Since F is relatively injective, there exists a G-mapw : B → F satisfying w v = ι u. Then setting w = r w we obtain aG-map with values in E and such that w v = u.

In the next theorem we finally obtain two more strong, relatively injectiveresolutions of G (and its Lie subgroups); they involve only smooth functionsand/or differential forms on a quotient manifold.

55

Theorem 3.4.7. Let G′ be a Lie subgroup of G and recall that M is themanifold G/H, with H compact. Then:

1. the following is a strong, relatively injective resolution of the trivialG′-module R

0 R C∞(M,R) ... C∞(Mk+1,R) ... ;ε d0 dk−1 dk

2. if M is contractible, another strong, relatively injective resolution isgiven by

0 R C∞(M,R) ... Ωk(M,R) ... .ε d0 dk−1 dk

Proof. Since we can identify C∞(Mk+1,R) with the subspace

C∞Hk+1(Gk+1,R) ⊆ C∞(Gk+1,R) ,

the first complex can be seen as a closed subcomplex of the strong resolutionmet in Corollary 3.4.2. The contracting homotopy that we constructed inthat context leaves this subcomplex invariant, thus proving that it is a strongresolution itself.The second complex is a strong resolution due to Proposition 3.4.4.Relative injectivity follows from Lemma 3.4.6.

The contractibility of M is given for example by Corollary 2.4.9 in thecase where G is semisimple and H is a maximal compact subgroup.

Remark 3.4.8. Theorem 3.4.7 shows that, if M is contractible, H∗c (G,R)can be computed via the complex

0 C∞(M,R)G Ω1(M,R)G Ω2(M,R)G ...d0 d1 d2

of G-invariant forms on M .Actually, if M has nonpositive curvature, we can find explicitly an extensionof the identity of R inducing an equivalence between the resolution given inPart 2 of Theorem 3.4.7 and the standard resolution. This remark is due toDupont ([Dup76]).For g0, ..., gk ∈ G, let ∆(g0, ..., gk) be the straight simplex in M (defined inRemark 1.4.8) with vertices g0, ..., gk.Now we can define V k : Ωk(M,R)→ C(Gk+1,R) as

V k(ω)(g0, ..., gk) :=

∫∆(g0,...,gk)

ω .

56

It is easy to check that V k is well-defined and continuous. Moreover, V k isa G-map since

V k(g · ω)(g0, ..., gk) =

∫∆(g0,...,gk)

(τ(g−1))∗ω =

=

∫∆(g−1g0,...,g−1gk)

ω = (g · V k(ω))(g0, ..., gk).

Moreover V ∗ is a chain map due to Stokes’ Theorem and it is clear that V ∗

extends the identity of R.Finally, observe that, if G′ is a Lie subgroup of G, composing each V k withthe restriction map

C(Gk+1,R)→ C(G′k+1,R) ,

we also obtain a G′-equivariant chain map

(Ω∗(M,R), d∗)→(C(G′∗+1,R), d∗

)extending idR.

The next result is usually known as Van Est’s Theorem and provides apowerful connection between the continuous cohomology of G and the spaceof G-invariant forms on a contractible quotient manifold. We will exploitthis fact mainly in the formulation given in Remarks 3.4.8 and 3.4.12.

Theorem 3.4.9. Let G be a connected Lie group with a compact subgroupH. If M = G/H is contractible and H is connected, we have

Hkc (G,R) ' Ωk

h(g/h) ' Ωk(M,R)G

for each k ≥ 0, where the elements of Ωkh(g/h) are the alternating, k-

multilinear maps f : (g/h)k → R satisfying the additional condition

k∑j=1

f(X1, ..., [Y,Xj ], ..., Xk) = 0

for each Y ∈ h. Moreover, a G-invariant form on M is always closed.

Proof. Theorem 3.4.7 shows thatH∗c (G,R) can be computed via the complex

0 C∞(M,R)G Ω1(M,R)G Ω2(M,R)G ... .d0 d1 d2

We will prove that this complex is isomorphic to the degenerate complex

0 Ω0h(g/h) Ω1

h(g/h) Ω2h(g/h) ... ,0 0 0

57

by defining maps φk : Ωkh(g/h)→ Ωk(M,R)G and ψk : Ωk(M,R)G → Ωk

h(g/h)and showing that they yield an isomorphism between the two complexes.First, fix an h-invariant metric on g (whose existence is guaranteed byLemma 1.3.5); setting p := h⊥, we obtain a decomposition g = h ⊕ p with[h, p] ⊆ p. Observe that, since H is connected, all the elements of AdG(H)are isometries of our given metric on g. In particular, if we extend it to aleft-invariant metric on G, it will be also invariant under right translationsby elements of H; this can be seen as in the proof of Theorem 2.4.3. In thefollowing discussion, we assume such a metric on G to be fixed.If ω is a G-invariant p-form onM , we can consider ωo ∈ Λp(p∗) ' Λp((g/h)∗),where o is the point H ∈ G/H = M . For Y ∈ h and t ∈ R, we can con-sider the diffeomorphism Ft = τ(exp(tY )) of M . Since Ft fixes o and ω isG-invariant, we have

d

dtωo (doFt(X1), ..., doFt(Xk))

∣∣t=0

= 0

for X1, ..., Xk ∈ p. Since the following diagram commutes

G

Cg // G

M

τ(g) //M

it is evident that doFt = AdG(exp(tY )) = exp(t · adg(Y )), under the identi-fication p ' ToM . Hence

0 =d

dtωo (doFt(X1), ..., doFt(Xk))

∣∣t=0

=k∑j=1

ωo(X1, ..., [Y,Xj ], ..., Xk)

for X1, ..., Xk ∈ p ' ToM and Y ∈ h. So ωo ∈ Ωph(g/h) and we can define

ψk(ω) := ωo.Conversely, if f ∈ Ωp

h(g/h), we can define φk(f) as the extension of f to aG-invariant form on M . This is possible if and only if τ(h)∗f = f for eachh ∈ H. It is sufficient to prove that this identity holds for the elements ofthe form h = exp(Y ), with Y ∈ h, since the image of the exponential mapcontains an open neighbourhood of the identity element and hence generatesall H. Define

F (t) := f(doFt(X1), ..., doFt(Xk)) = (F ∗t f)(X1, ..., Xk)

for X1, ..., Xk ∈ p; we need to show that F is constant. The previous discus-sion shows that F ′(0) = 0, but since

F (t+ t0) = (F ∗t F∗t0f)(X1, ..., Xk)

58

and F ∗t0f lies in Ωph(g/h) as well, we obtain F ′(t0) = 0 for each t0.

Now we only have to prove that φk and ψk are chain maps because it is clearthat their compositions are the identity. Clearly it is sufficient to constructvector fields X1, ..., Xk+1 near o having assigned values (X1)o, ..., (Xk+1)o ato and satisfying, for each G-invariant k-form ω:

Xi

(ω(X1, ..., Xi, ..., Xk+1)

)= 0, ∀i = 1, ..., k + 1,

[Xi, Xj ]o = 0, ∀i, j.

Choose a smooth local section S of π : G→M near o. The vectors

(X1)o, ..., (Xk+1)o ∈ ToM ' p ⊆ g

can be extended to left-invariant vector fields X1, ..., Xk+1 on G; their re-strictions to the image of S can be pushed forward via π to vector fields onM near o. We define X1, ..., Xk+1 this way.It is easy to check that ω(X1, ..., Xi, ..., Xk+1) is constant near o, so that thefirst requirement is satisfied. Moreover, ([Xi, Xj ])o lies in h for each i, j, so([Xi, Xj ])o = 0.Since we have obtained an isomorphism between the degenerate complexΩ∗h(g/h) and the complex Ω∗(M,R)G, we conclude that the latter complexmust be degenerate as well; this means precisely that invariant forms on Mare closed.

Corollary 3.4.10. The real continuous cohomology of a connected compactLie group G is trivial.

Proof. Follows from Theorem 3.4.9 by choosing H = G.

Corollary 3.4.11. In the setting of Theorem 3.4.9, we have:

Hnc (G,R) ' R ,

Hkc (G,R) = 0 for each k > n,

where n = dim M and the other cohomology spaces are finite-dimensional.The hypotheses of Theorem 3.4.9 are always satisfied if G is semisimple andH is maximal compact.

Proof. From Theorem 3.4.9:

dim Hhc (G,R) = dim Ωh

h(g/h) ≤ dim Λh((g/h)∗) < +∞, ∀h ∈ N,

dim Hnc (G,R) = dim Ωn

h (g/h) ≤ dim Λn((g/h)∗) = 1,

dim Hnc (G,R) = dim Ωn(M,R)G ≥ 1,

59

where the last inequality is due to the fact that M admits G-invariant Rie-mannian metrics (see Section 2.1) and the volume form associated with anyof these metrics is a nontrivial element of Ωn(M,R)G.The fact that Hk

c (G,R) = 0 for k > n follows from Λk((g/h)∗) = 0.Corollary 2.4.9 guarantees that M is contractible in the mentioned case; theconnectedness of H follows from Proposition 2.4.8.

Remark 3.4.12. When (G,H) is a Riemannian symmetric pair of noncom-pact type (or, more generally, when G,H are connected Lie groups withG/H contractible and supporting a G-invariant metric of nonpositive cur-vature), we can exploit Remark 3.4.8 to construct a cocycle in C(Gn+1,R)G

that represents a cohomology class generating Hnc (G,R).

Endowing M with a G-invariant metric, we obtain a nonpositively curvedspace, so that the straight simplices ∆(g0, ..., gn) are well-defined for allg0, ..., gn ∈ G; moreover, the associated volume form volM generates thespace Ωn(M,R)G. Hence we obtain the cocycle

v(g0, ..., gn) = volM (∆(g0, ..., gn)).

We remark that volM (σ) represents the integral of volM over the simplex σand hence is negative if σ is negatively oriented.

Remark 3.4.13. It is possible to define a notion of Lie algebra cohomology,concept originally due to Chevalley and Eilenberg (see [CE48]). Given Liealgebras h ⊆ g (usually one requires h to be reductive in g, condition that isalways verified for compactly embedded subalgebras), it is possible to definethe space Hk(g, h, E), where E is a (possibly infinite-dimensional) represen-tation of g.We will not develop this theory, even though it can be approached by mim-icking the content of Section 3.1, the main complication being the proof ofthe existence of a resolution of E having the required properties (see [Gui80]for more details).Note however that Hk(g, h,R) can be computed via the complex Ω∗h(g/h)introduced in Theorem 3.4.9.

60

Chapter 4

Bounded Cohomology

The main characters in this chapter will be continuous bounded cohomol-ogy spaces for topological groups, bounded cohomology spaces for manifoldsand, of course, the notion of simplicial volume. We will define these conceptsin Sections 4.1 and 4.2, then explore their mutual relationships, mainly inSections 4.2 and 4.5.More precisely, we will see that bounded cohomology spaces are endowedwith canonical seminorms and that the problem of computing the simpli-cial volume of a manifold M can be reformulated as a question about theseminorm of particular classes in the bounded cohomology of M (Proposi-tion 4.2.7). Then we shall show that the bounded cohomology of a manifold isactually the bounded cohomology of its fundamental group (Theorem 4.2.9),i.e of a discrete subgroup of I(M), where M is the universal covering of M ,assuming that M is endowed with a Riemannian metric.Finally, in Section 4.5 we will provide a connection between the continuousbounded cohomology of a topological group and that of certain discrete sub-groups (Theorem 4.5.3). In the light of the preceding results this is actuallya relationship between the bounded cohomology of M and that of I(M)(Corollary 4.5.6); we will exploit it in Chapter 5 to obtain a characterisationof the simplicial volume of locally symmetric spaces of noncompact type interms of the seminorm of certain continuous bounded cohomology classes ofI(M), result that we will then employ in Chapter 6 to compute the simplicialvolume of manifolds covered by H2 ×H2.

4.1 Continuous bounded cohomology

We will introduce here new notions of strong map and relatively injectiveG-module. We will work in the category of Banach spaces and Banach G-modules, whereas we were considering the categories of topological vectorspaces and Fréchet G-modules in Chapter 3.Throughout this chapter, G will be a locally compact, Hausdorff, second-

61

countable topological group and E,F will be Banach spaces.

Definition 4.1.1. An injective, continuous, linear map f : E → F is saidto be strong if it admits a linear, 1-Lipschitz left inverse s. In general, f isstrong if Kerf → E and f : E/Kerf → F are.A Banach G-module is a Banach space E equipped with a left action of Gby linear isometries (without any additional continuity assumption, unlikeDefinition 1.5.11). We say that E is a continuous Banach G-module if themap

G× E → E

(g, e)→ g · e

is continuous; this is equivalent to requiring simply a condition of separatecontinuity thanks to Theorem 1.5.14. The continuity requirement was au-tomatically included in the definition of Fréchet module that we gave inSection 1.5; we decided to separate it from the concept of Banach modulesince we will be forced to consider G-modules which are not continuous.A G-map is a continuous, linear map between Banach G-modules that isG-equivariant.A Banach G-module E is relatively injective if, whenever continuous BanachG-modules A,B are given, jointly with G-maps u : A → E and v : A → B,with v strong and injective, there exists a G-map w : B → E satisfying‖w‖ ≤ ‖u‖ and causing the following diagram to commute:

A B

E .

u

v

w

A resolution of a Banach G-module E is an exact complex of Banach G-modules and G-maps

0 // Eε // E0 d0 // E1 d1 // ...

dk−1// Ek

dk // ... .

A resolution is said to be strong if all the involved maps are strong andrelatively injective if each Ek, k ≥ 0, is relatively injective. Moreover, aresolution of a continuous G-module E will be called continuous if each Ek

is a continuous G-module.A contracting homotopy for a resolution is a family of linear maps sk : Ek →Ek−1, for k ≥ 0 (where E−1 has to be understood as E), satisfying

dk−1sk + sk+1dk = id

and ‖sk‖ ≤ 1 for each k ≥ 0.

We can construct several Banach G-modules, as the next example shows.

62

Example 4.1.2. Let X be a set on which G acts on the left and let E be aBanach G-module. Then, for a function f : X → E we can define

‖f‖∞ := supx∈X‖f(x)‖ ∈ [0,+∞]

and observe that the space

Cb(X,E) := f : X → E | ‖f‖∞ <∞

is a Banach space when endowed with ‖ · ‖∞.Moreover, Cb(X,E) becomes a Banach G-module (in general not continuous)when we define the following action of G:

(g · f)(x) = g ·(f(g−1 · x)

).

Observe that the maps dk : Cb(Xk+1, E)→ Cb(X

k+2, E),

(dkf)(x0, ..., xk+1) :=

k+1∑j=0

(−1)j · f(x0, ..., xj , ..., xk+1) ,

are G-maps when we consider the diagonal action of G on Xk+1, Xk+2.If X is endowed with a topology and G acts on X by homeomorphisms, thesubspace of continuous and bounded functions

Cb(X,E) ⊆ Cb(X,E)

is closed and G-invariant; in particular, it is a Banach G-module itself. Inthe rest of this section, we will be especially interested in G-modules of theform Cb(Gk+1, E), k ≥ 0.

We now proceed to proving in the present context a few statements anal-ogous to those in Section 3.1. We will show that the cohomology of thecomplex of invariants associated to a continuous, strong, relatively injectiveresolution of a continuous Banach G-module does not depend on the chosenresolution (compare with Corollary 3.1.7). Then we will construct a “stan-dard resolution” with these properties for each continuous Banach G-moduleE and we will define the spaces H∗cb(G,E) as the cohomology spaces of theassociated complex of invariants.Our approach is based on Monod’s book ([Mon01]) but differs from it in thatwe will only consider cohomology with coefficients in continuous G-modulesand restrict our attention to their continuous resolutions. This considerablysimplifies the required arguments and does not impoverish the theory aswe recover Monod’s definition for non-continuous G-modules E by settingH∗cb(G,E) := H∗cb(G, CE) (see Proposition 4.1.4 below for the definition ofthe module CE). Note however that our definition of strong resolution differsfrom that in [Mon01].

63

Proposition 4.1.3. A resolution of a Banach G-module is strong if andonly if it admits a contracting homotopy.

Proof. Analogous to Proposition 3.1.3.

Since we will sometimes need to consider non-continuous Banach G-modules, we will need the following result.

Proposition 4.1.4. Let E be a Banach G-module. Then:

1. the subspace

CE := e ∈ E | g 7→ g · e is continuous

is a continuous Banach G-module containing EG and (CE)G = EG;

2. a G-map E → F sends CE into CF ;

3. E is relatively injective if and only if CE is.

Proof. The subspace CE is closed in E. Indeed, if en → e and en ∈ CE wecan prove that, whenever gk → g in G, we have gk · e→ g · e. Sure enough,given ε > 0, choose n with ‖e− en‖ < ε; then:

‖gk · e− g · e‖ ≤ ‖gk · e− gk · en‖+ ‖gk · en − g · en‖+ ‖g · en − g · e‖ =

= 2‖e− en‖+ ‖gk · en − g · en‖ < 2ε+ ‖gk · en − g · en‖

and ‖gk · en− g · en‖ converges to 0 as k goes to infinity, so that, if k is largeenough, one has ‖gk · e− g · e‖ < 3ε.Hence CE is a Banach G-module and, by Proposition 1.5.14, it is continuous.The rest of part 1 is trivial, as well as part 2. Part 3 then easily follows.

Proposition 4.1.5. Let f : E → F be a G-map between Banach G-modulesand let (E,E∗, d∗), (F, F ∗, δ∗) be resolutions of E and F . Assume that(E,E∗, d∗) is strong and (F, F ∗, δ∗) is relatively injective.If, in addition, E is a continuous G-module and (E,E∗, d∗) is a continuousresolution, then f can be extended to a chain map made of G-maps. Thisextension is unique up to homotopies made of G-maps.

Proof. Simply reproduce the proof of Proposition 3.1.5.

Proposition 4.1.5 also provides a version of Corollary 3.1.7 in the presentcontext.

Definition 4.1.6. The k-th continuous bounded cohomology space of G withcoefficients in a continuous Banach G-module E is the k-th cohomologymodule of the complex

0 // (E0)Gd0 // (E1)G

d1 // ...dk−1// (Ek)G

dk // ... .

64

associated with any continuous, strong, relatively injective resolution of E:

0 // Eε // E0 d0 // E1 d1 // ...

dk−1// Ek

dk // ... .

We will denote it by Hkcb(G,E); thanks to Proposition 4.1.5, it does not

depend on the chosen resolution of E.We also define Zkb := Ker dk and Bk

b := Im dk−1 whenever the resolution isclear from the context; so Hk

cb(G,E) = Zkb /Bkb .

If E is not a continuous G-module, we can set Hkcb(G,E) := Hk

cb(G, CE).

Remark 4.1.7. The spaceHkcb(G,E) does not necessarily inherit a structure

of Banach space, as the coboundary maps in the complex of invariants couldnot have a closed image. We shall see later, however, that these spaces arealways equipped with a canonical seminorm.

We now proceed to constructing a continuous, strong, relatively injec-tive resolution for each continuous Banach G-module E, thus showing thatthe spaces H∗cb(G,E) are well-defined. We will refer to it as the standardresolution, but it should be noted that this terminology differs from that inMonod’s book ([Mon01]).

Lemma 4.1.8. The Banach G-modules Cb(Gk+1, E) are relatively injective.

Proof. Let A,B, u, v be given as in Definition 4.1.1 and s a left inverse for vwith ‖s‖ ≤ 1. Define

(w(b)) (g0, ..., gk) := u(g0 · s(g−1

0 · b))

(g0, ..., gk) ,

for each b ∈ B and g0, ..., gk ∈ G. The function w(b) is continuous since Aand B are continuous G-modules. Moreover:

‖(w(b))(g0, ..., gk)‖ ≤∥∥u (g0 · s(g−1

0 · b))∥∥∞ ≤ ‖u‖ ·

∥∥s(g−10 · b))

∥∥ ≤ ‖u‖ · ‖b‖,so w(b) is bounded and we have ‖w‖ ≤ ‖u‖.The proof of the fact that w v = u and that w is equivariant is identical tothat given for Proposition 3.1.12.

Proposition 4.1.9. The augmented complex(E, C(Cb(G∗+1, E)), d∗

)is a

continuous, strong, relatively injective resolution of E, for each continuousBanach G-module E.

Proof. Thanks to Lemma 4.1.8 and Proposition 4.1.4, we only have to provethat this is a strong resolution; we will construct a contracting homotopy.Let µ be a left Haar measure on G and h : G → [0,+∞) a compactly sup-ported, continuous function with

∫G h(g−1) dµ(g) = 1.

Define, for f ∈ C(Cb(Gk+1, E)) and k ≥ 1,

(skf)(g0, ..., gk−1) :=

∫Gh(g−1)f(g, g0, ..., gk−1) dµ(g),

65

and (s0f) =∫G h(g−1)f(g) dµ(g).

Proposition 1.5.19 guarantees that skf ∈ Cb(Gk, E) and that ‖sk‖ ≤ 1.To check that skf lies in C(Cb(Gk, E)), consider a sequence gn → g in G; wehave to prove that gn · (skf)→ g · (skf). Without loss of generality, we canassume that g = e. Now:

‖gn · (skf)− skf‖∞ ≤ ‖gn · (skf)− sk(gn · f)‖∞ + ‖sk(gn · f − f)‖∞

and the second term tends to zero as n tends to infinity. Denoting by τ(g)the isometry of E given by g ∈ G and recalling that Lkg is the diffeomorphismof Gk that is Lg on each factor:

gn · (skf)− sk(gn · f) = τ(gn) (skf) Lkg−1n− sk

(τ(gn) f Lk+1

g−1n

)=

= τ(gn) (

(skf) Lkg−1n− sk

(f Lk+1

g−1n

)).

Hence

‖gn · (skf)− sk(gn · f)‖∞ =∥∥∥(skf) Lk

g−1n− sk

(f Lk+1

g−1n

)∥∥∥∞

=

= supg0,...,gk−1

∥∥∥∥∥∫Gh(g−1)

[f(g, g−1

n g0, ..., g−1n gk−1)+

− f(g−1n g, g−1

n g0, ..., g−1n gk−1)

]dµ(g)

∥∥∥∥∥ =

= supg0,...,gk−1

∥∥∥∥∫G

(h(g−1 − h(g−1g−1n ))f(g, g−1

n g0, ..., g−1n gk−1) dµ(g)

∥∥∥∥ ≤≤ ‖f‖∞ ·

∫G

∣∣h(g−1)− h(g−1g−1n )∣∣ dµ(g),

which tends to zero by standard results in the theory of Lebesgue integration.Finally, we check that dk−1sk + sk+1dk = id for each k ≥ 0 (where d−1 = ε):

(dk−1skf)(g0, ..., gk) =

k∑j=0

(−1)j ·∫Gh(g−1)f(g, g0, ..., gj , ..., gk) dµ(g) ,

(sk+1dkf)(g0, ..., gk) =

=

∫Gh(g−1)

f(g0, ..., gk)−k∑j=0

(−1)jf(g, g0, ..., gj , ..., gk)

dµ(g) =

= f(g0, ..., gk)− (dk−1skf)(g0, ..., gk) .

66

Corollary 4.1.10. The continuous bounded cohomology of G with coeffi-cients in a continuous Banach G-module E can be computed as the cohomol-ogy of the complex

0 Cb(G,E)G Cb(G2, E)G Cb(G3, E)G ... .d0 d1 d2

Proof. By Proposition 4.1.9, H∗cb(G,E) can be computed as the cohomologyof the complex C(Cb(G∗+1, E)). The result then follows from the observationthat C(Cb(G∗+1, E))G = Cb(G∗+1, E)G.

Remark 4.1.11. If G = Γ is a countable, discrete group, every isometricaction of Γ on a Banach space E yields a continuous Γ-module (see Propo-sition 1.5.14). In particular:

C(Cb(Γk+1, E)) = Cb(Γk+1, E) = Ckb (Γ, E) := f : Γk+1 → E | f is bounded.

Hence H∗cb(Γ, E) is simply the cohomology of the complex of bounded, Γ-invariant, E-valued functions:

0 C0b (Γ, E)Γ C1

b (Γ, E)Γ ... Ckb (Γ, E)Γ ... .d0 d1 dk−1 dk

For discrete groups, we will simply refer to these spaces as the boundedcohomology spaces and denote them by H∗b (Γ, E).

The standard resolution allows us to define a seminorm on each spaceHkcb(G,E) as

‖α‖ := inf ‖f‖∞ | f ∈ Cb(Gk+1, E)G, [f ] = α.

The same construction, associates a seminorm on Hkcb(G,E) to each contin-

uous, strong, relatively injective resolution (E,E∗, δ∗) of E:

‖α‖′ := inf ‖e‖ | e ∈ (Ek)G, [e] = α.

However, the norm ‖ · ‖ induced by the standard resolution is canonical,being the minimal such seminorm, as the next result shows.

Theorem 4.1.12. Let (E,E∗, δ∗) be a continuous, strong, relatively injectiveresolution of the continuous Banach G-module E. Then there are G-mapsLk : Ek → Cb(Gk+1, E) extending id : E → E and satisfying ‖Lk‖ ≤ 1:

0 E E0 E1 E2 ...

0 E Cb(G,E) Cb(G2, E) Cb(G3, E) ... .

η

id L0

δ0

L1

δ1 δ2

L2

ε d0 d1 d2

Moreover, each Lk actually ranges in C(Cb(Gk+1, E)), since the Ek are con-tinuous G-modules.

67

Proof. Let s∗ be a contracting homotopy for the resolution (E,E∗, δ∗), setL−1 = idE and define inductively, for k ≥ 0:

Lk(e)(g0, ..., gk) := Lk−1(g0 · sk(g−1

0 · e))

(g1, ..., gk) .

It is clear that Lke is a continuous function and that ‖Lk‖ ≤ 1, since∥∥∥Lk(e)(g0, ..., gk)∥∥∥ ≤ ∥∥∥g0 · sk(g−1

0 · e)∥∥∥ =

∥∥∥sk(g−10 · e)

∥∥∥ ≤ ‖e‖ .Furthermore, Lk is equivariant; indeed, using the inductive hypothesis:(g · Lk(e)

)(g0, ..., gk) = g·

(Lk−1

(g−1g0 · sk(g−1

0 g · e))

(g−1g1, ..., g−1gk)

)=

= Lk−1(g0 · sk(g−1

0 g · e))

(g1, ..., gk) = Lk(g · e)(g0, ..., gk) .

Finally, again using the inductive hypothesis, the Lkk≥0 form a chain map:(Lkδk−1e

)(g0, ..., gk) = Lk−1

(g0 · sk(g−1

0 · δk−1e)

)(g1, ..., gk) =

= Lk−1(g0 · skδk−1(g−1

0 e))

(g1, ..., gk) =

= Lk−1(g0 · (id− δk−2sk−1)(g−1

0 e))

(g1, ..., gk) =

= Lk−1(e)(g1, ..., gk)− Lk−1δk−2(g0 · sk−1(g−1

0 e))

(g1, ..., gk) =

= Lk−1(e)(g1, ..., gk)− dk−2Lk−2(g0 · sk−1(g−1

0 e))

(g1, ..., gk) =

= Lk−1(e)(g1, ..., gk)+k∑j=1

(−1)j ·Lk−2(g0 · sk−1(g−1

0 · e))

(g1, ..., gj , ..., gk) =

=(dk−1Lk−1e

)(g0, ..., gk) .

The next result gives some information regarding when a seminorm isactually a norm.

Proposition 4.1.13. A seminorm ‖ · ‖′ on Hkcb(G,E) is a norm if and only

if the image Bkb of the map dk−1 : Ek−1 → Ek is closed in Ek. This is always

the case if Hkcb(G,E) is finite-dimensional.

Proof. A class α = [c] ∈ Hkcb(G,E) has vanishing seminorm if and only if

c + Bkb accumulates on 0 ∈ Ek, that is if and only if c lies in the closure of

Bkb in Ek; this proves the first part of the proposition. The second part is

just Lemma 3.1.9.

68

Remark 4.1.14. If ‖ · ‖′ is a norm, then(Hkcb(G,E), ‖ · ‖′

)is a Banach

space and the associated topology is the quotient topology of Zkb /Bkb , see

Lemma 1.5.5.

Remark 4.1.15. The functoriality results of Remark 3.1.10 and Section 3.2hold also in this context, with essentially the same proofs (considering themodules C(Cb(G∗+1, E)) instead of C(G∗+1, E) and noticing that f∗E is acontinuous H-module whenever E is a continuous G-module).Observe in addition that the map

f∗ : Hkcb(G,E)→ Hk

cb(H, f∗E)

is 1-Lipschitz with respect to the canonical seminorms.

We conclude this section by observing that, for a continuous BanachG-module, we have continuous inclusions

C(Cb(Gk+1, E)) → Cb(Gk+1, E) → C(Gk+1, E)

which induce comparison maps

ιk : Hkcb(G,E)→ Hk

c (G,E) .

These maps are continuous with respect to the quotient topologies on thecohomology spaces.

Lemma 4.1.16. Let E be a continuous Banach G-module and (E,E∗, d∗)a continuous, strong, relatively injective resolution. Let F be a Fréchet G-module and let (F, F ∗, d∗) be a strong, relatively injective resolution of F .Suppose that we have G-equivariant, continuous chain maps

f∗, g∗ : (E,E∗, d∗)→ (F, F ∗, d∗)

extending the same continuous, G-equivariant, linear map h : E → F .Then, f∗ and g∗ are homotopic via a continuous, G-equivariant homotopy.

Proof. The resolution (E,E∗, d∗) can be viewed as a strong, relatively in-jective resolution of E in the category of Fréchet G-modules, i.e. in thesense of Definitions 3.1.1 and 3.1.4. Then, the lemma follows from Proposi-tion 3.1.5.

Proposition 4.1.17. Let E be a continuous Banach G-module and (E,E∗, d∗)a continuous, strong, relatively injective resolution. Moreover, let (E,F ∗, d∗)be a strong, relatively injective resolution of E in the category of Fréchet G-modules, i.e. in the sense of Definitions 3.1.1 and 3.1.4.Let f∗ : (E,E∗, d∗)→ (E,F ∗, d∗) be a G-equivariant, continuous chain mapextending id : E → E.Then the induced morphisms f∗ : Hk

cb(G,E) → Hkc (G,E) are precisely the

comparison maps ιk, k ≥ 0.

69

Proof. Define g∗ : (E,E∗, d∗)→ (E,F ∗, d∗) by composing the map

C(Cb(G∗+1, E)) → Cb(G∗+1, E) → C(G∗+1, E)

on the right with a morphism (E,E∗, d∗) → (E, C(Cb(G∗+1, E)), d∗) ex-tending id : E → E and on the left with a morphism (E, C(G∗+1, E), d∗)→(E,F ∗, d∗) extending id : E → E; the existence of these morphisms is guar-anteed by Propositions 3.1.5 and 4.1.5. Conclude by applying the previouslemma.

Finally, the canonical seminorm on Hkcb(G,E) allows us to define a semi-

norm also on Hkc (G,E), possibly taking the value +∞.

We can set ‖α‖ := +∞ for all those classes α that do not lie in the image ofthe comparison map ιk and

‖α‖ := inf‖β‖ | β ∈ Hk

cb(G,E), ιk(β) = α< +∞

in all the other cases.

4.2 Simplicial volume and bounded cohomology

It is possible to define bounded cohomology also for topological spaces; as weshall see, the construction is very similar to the one for singular cohomology.We will be concerned only with real coefficients.Let X be a topological space and f : Ck(X,R)→ R a real singular k-cochain;we define its `∞-norm as

‖f‖∞ := sup |f(σ)| | σ is a singular k-simplex ∈ [0,+∞] .

We can consider the subspaces

Ckb (X,R) := f ∈ Ck(X,R) | ‖f‖∞ < +∞

and observe that they form a subcomplex of the usual complex (C∗(X,R), d∗)computing the singular cohomology of X.

Definition 4.2.1. The k-th cohomology space of the complex (C∗b (X,R), d∗)will be denoted by Hk

b (X,R) and called the k-th bounded cohomology spaceof X.

Moreover, we define the spaces Zkb (X,R) := Zk(X,R) ∩ Ckb (X,R) andBkb (X,R) := dk−1(Ck−1

b (X,R)).Just as in Section 4.1, the `∞-norm on Ckb (X,R) induces a seminorm onHkb (X,R), which we shall simply denote by ‖ · ‖.

The inclusion of complexes C∗b (X,R) → C∗(X,R) induces comparison mor-phisms

ιk : Hkb (X,R)→ Hk(X,R) .

70

We remark that there are examples of topological spaces for which the com-parison morphisms are not injective (e.g. when X is a surface of genus ≥ 2and k = 2, 3, see [BS84] and [Som97]) or not surjective (e.g. when X is ann-torus and k = 1, n, see Corollary 4.4.16).The comparison morphisms allow us to define a seminorm also on Hk(X,R),taking values in [0,+∞]. Indeed, we can define ‖α‖ = +∞ for all thoseclasses α that do not lie in the image of ιk and

‖α‖ := inf‖β‖ | β ∈ Hk

b (X,R), ιk(β) = α< +∞

in all the other cases.We now at last introduce formally the concept of simplicial volume of amanifold. As we shall see, it is strictly related to the seminorm which wehave just endowed bounded cohomology spaces with.

Definition 4.2.2. LetMn be a connected, closed, oriented topological man-ifold with fundamental class [M ] ∈ Hn(M,R). The simplicial volume (orGromov norm) of M is:

‖M‖ := inf ‖c‖1 | c ∈ Zn(M,R), [c] = [M ] ,

where the `1-norm of a singular cycle c =∑

α λα · σα (with σα 6= σβ forα 6= β) is defined as

‖c‖1 :=∑α

|λα| .

We will denote by ωM the dual fundamental class ofM , i.e. the only elementof Hn(M,R) ' R satisfying ωM ([M ]) = 1.Observe that the `1-norm on the spaces Ck(M,R) actually induces a semi-norm on each homology space Hk(M,R); we will call it Gromov norm aswell.

The next two lemmas are useful properties of the simplicial volume as atopological invariant.

Lemma 4.2.3. Let f : M → N be a continuous map between connected,closed, oriented manifolds and set d = deg f . Then:

d · ‖N‖ ≤ ‖M‖.

Proof. If c is a cycle representing [M ], the cycle f∗(c) represents d · [N ] and‖f∗(c)‖1 ≤ ‖c‖1. Hence:

d · ‖N‖ = ‖f∗([M ])‖ ≤ ‖f∗(c)‖1 ≤ ‖c‖1,

and, taking the infimum over c, we get d · ‖N‖ ≤ ‖M‖.

71

Corollary 4.2.4. If ‖M‖ 6= 0, every continuous map f : M → M satisfies| deg f | ≤ 1.

Example 4.2.5. The sphere Sn and the torus Tn have vanishing simplicialvolume for each n ≥ 1 since they have self-maps with arbitrarily large degree.

Lemma 4.2.6. If f : M → N is a finite covering map, then

(deg f) · ‖N‖ = ‖M‖ .

Proof. We already know that (deg f) · ‖N‖ ≤ ‖M‖ from Lemma 4.2.3. Letc =

∑α λα · σα be a cycle on N representing the fundamental class; for each

α, let σ1α, ..., σ

deg fα be the distinct lifts of σα to a singular simplex inM . The

chainc =

∑α

λα · (σ1α + ...+ σdeg f

α )

is a cycle on M and it is clear that it represents [M ]. Since its norm is‖c‖1 = (deg f) · ‖c‖1, the arbitrarity of c yields (deg f) · ‖N‖ ≥ ‖M‖.

The next result is the main reason why we have introduced boundedcohomology in order to study the simplicial volume of manifolds. It is akey-point in the computation of the simplicial volume of manifolds coveredby H2 ×H2 which we will carry out in Chapter 6.

Proposition 4.2.7. LetMn be a connected, closed, oriented manifold. Then:

‖M‖ =1

‖ωM‖.

Proof. Let c =∑

α λα · σα be a cycle representing [M ] and f a cocyclerepresenting ωM . We have

1 = f(c) =

∣∣∣∣∣∑α

λα · f(σα)

∣∣∣∣∣ ≤∑α

|λα| · |f(σα)| ≤ ‖c‖1 · ‖f‖∞,

and taking the infimum over all f and c we obtain: 1 ≤ ‖M‖ · ‖ωM‖.If ‖M‖ = 0, it must be ‖ωM‖ = +∞ and the proposition is proven. So letus assume that ‖M‖ > 0; in this case, we surely have ‖M‖ ≥ 1/‖ωM‖ andit must be ‖ωM‖ < +∞.We will construct h ∈ Znb (M,R) with h(c) = 1 (hence ιn([h]) = ωM ) and‖h‖∞ = 1/‖M‖; this will complete the proof.Define h(c) = 1 and h|Bn(M,R) ≡ 0 (where Bn(M,R) denotes the subspace ofboundaries in Cn(M,R)). The resulting linear functional h : Rc ⊕ Bn → Ris continuous with respect to ‖ · ‖1:

‖h‖ = supλ∈Rb∈Bn

|h(λc+ b)|‖λc+ b‖1

= supλ∈Rb∈Bn

|λ|‖λc+ b‖1

= supb∈Bn

1

‖c+ b‖1=

1

‖M‖< +∞ .

72

So the Hahn-Banach Theorem grants us an extension h : Cn(M,R) → Rwith ‖h‖∞ = 1/‖M‖; since h vanishes on Bn(M,R), it is a cocycle and, byconstruction, we also have h(c) = 1.

Corollary 4.2.8. Let Mm, Nn be connected, closed, orientable manifolds.Then:

‖M‖ · ‖N‖ ≤ ‖M ×N‖ .

Proof. Fix orientations on M and N and endow M × N with the inducedorientation; let πM , πN be the projections ofM×N onto its factors. Observethat

ωM×N = (π∗MωM ) ` (π∗NωN ) .

Indeed:

〈(π∗MωM ) ` (π∗NωN ), [M ×N ]〉 = 〈ωM , [M ]〉 · 〈ωN , [N ]〉 = 1 ;

see for example the end of §3.B in [Hat02] for a proof of the first equality.If f ∈ Cmb (M,R) and g ∈ Cnb (N,R) represent ωM and ωN respectively, then:

‖ωM×N‖ ≤ ‖π∗Mf ` π∗Ng‖∞ ≤ ‖f‖∞ · ‖g‖∞

and, by the arbitrarity of f and g, we obtain ‖ωM×N‖ ≤ ‖ωM‖ · ‖ωN‖. Weconclude by Proposition 4.2.7.

We conclude this section with the following essential result. It providesa useful connection between cohomology and bounded cohomology of topo-logical spaces and the same concepts for discrete groups.

Theorem 4.2.9. Let M be a simply connected, nonpositively curved Rie-mannian manifold and Γ < I(M) a discrete subgroup acting freely on M .Then H∗(Γ\M,R) ' H∗(Γ,R) and H∗b (Γ\M,R) ' H∗b (Γ,R) and these iso-morphisms are isometric.

Proof. Consider the complex

... R[Γk+1] R[Γk] ... R[Γ2] R[Γ] 0 ,∂k+1 ∂k ∂k−1 ∂2 ∂1

where each ∂k is the linear map defined by

∂k([γ0, ..., γk]) =

k∑j=0

(−1)j · [γ0, ..., γj , ..., γk] ,

and Γ acts on each R[Γk+1] by

γ · [γ0, ..., γk] = [γγ0, ..., γγk] .

73

Fix x ∈ M and define a linear map:

φk : R[Γk+1]→ Ck(M,R)

[γ0, ..., γk] 7→ ∆(γ0 · x, ..., γk · x)

where ∆(γ0 · x, ..., γk · x) denotes a straight simplex in M , according toDefinition 1.4.8. It is easy to check that this defines a Γ-equivariant chainmap φ∗ : R[Γ∗+1]→ C∗(M,R).Now choose some subset ∆ ⊆ M containing x and such that the coveringprojection M → Γ\M maps it bijectively to the quotient; let f∆ : M → Γ

be the only function with the property that x ∈ f∆(x) ·∆, ∀x ∈ M .Now we can define the linear map:

ψk : Ck(M,R)→ R[Γk+1]

σ 7→ [f∆(σ0), ..., f∆(σk)]

where σ0, ..., σk are the vertices of the singular simplex σ. Again, it is clearthat ψ∗ is a chain map and it is Γ-equivariant because

f∆(γ · x) = γ · f∆(x), ∀γ ∈ Γ, ∀x ∈ M.

We have ψk φk = id since f∆(γ · x) = γ. Moreover, φk ψk is homotopic tothe identity.Indeed, we can construct Γ-equivariant maps Tk : Ck(M,R) → Ck+1(M,R)

realising this homotopy: if Hσ : ∆k × [0, 1] → M is the homotopy from σ

to (φk ψk)(σ) obtained by following the geodesics of M , it is sufficient todefine T (σ) = (Hσ)∗(P ) where P is the sum of (k+1)-simplices in ∆k× [0, 1]given by the usual triangulation of the prism. It is easy to check that T∗ isindeed the required homotopy and that it is Γ-equivariant.Observe that, considering the `1-norms on R[Γk+1] and Ck(M,R), we have‖φk‖ = 1, ‖ψk‖ = 1 and ‖Tk‖ = k + 1.Dualising the complexes, we obtain maps

(φk)∗ : Ck(M,R)→ Ck(Γ,R)

(ψk)∗ : Ck(Γ,R)→ Ck(M,R)

which, being Γ-equivariant, define a homotopy equivalence between the com-plexes

(C∗(Γ,R)Γ, d∗

)and

(C∗(M,R)Γ, d∗

)'(C∗(Γ\M,R), d∗

). This

proves that H∗(Γ\M,R) ' H∗(Γ,R).Furthermore, since φk, ψk, Tk have finite `1-norm, we also have:

(φk)∗ : Ckb (M,R)→ Ckb (Γ,R)

(ψk)∗ : Ckb (Γ,R)→ Ckb (M,R)

74

where the chain map (ψk)∗ (φk)

∗ is homotopic to the identity and moreover‖(φk)∗‖ ≤ 1, ‖(ψk)∗‖ ≤ 1. This implies that H∗b (Γ\M,R) ' H∗b (Γ,R)isometrically.The fact that also the isomorphism H∗(Γ\M,R) ' H∗(Γ,R) is isometricfollows from the commutativity of

Hk(Γ\M,R) Hk(Γ,R)

Hkb (Γ\M,R) Hk

b (Γ,R) .

'

ιk

'

ιk

Remark 4.2.10. More generally, the previous theorem holds also if M isan arbitrary contractible topological space and Γ is the group of coveringautomorphisms of some covering map M → M . The proof is essentiallythe same, the only problem being that the maps ψ∗ and T∗ have to be con-structed inductively because of the lack of a canonical procedure to constructa simplex with given vertices.

Remark 4.2.11. Even more generally, if M is any topological space withthe homotopy type of an at most countable CW-complex and Γ is the groupof covering automorphisms of some covering map M →M , we can still definethe chain maps φ∗ and ψ∗ and obtain a commutative diagram:

Hk(M,R) Hk(Γ,R)(ψk)∗oo

Hkb (M,R)

ιk

OO

Hkb (Γ,R) .

ιk

OO

(ψk)∗oo

In this very general context, the map (ψk)∗ : Hk(Γ,R) → Hk(M,R) will

not always be an isomorphism. However, surprisingly enough, the map(ψk)

∗ : Hkb (Γ,R) → Hk

b (M,R) is always an isometric isomorphism. Thiswas originally proven by Gromov in [Gro82]; see also [Iva85] for a more de-tailed proof.In particular, if Mn is a closed (non-aspherical) manifold and Hn(Γ,R) = 0,we deduce from the commutativity of the diagram above that ωM cannot liein the image of ιn (which must be trivial); hence ‖M‖ = 0.

Example 4.2.12. As a consequence of the previous remark, all simply con-nected, closed manifolds (of dimension at least 1) have vanishing simplicialvolume.

75

4.3 Smooth homology and cohomology

In this section we recall a few facts on smooth singular homology and smoothsingular cohomology. Their main interest lies in the fact that they allow usto integrate differential forms on homology classes. In particular, we willneed these theories to prove the multiplicativity of the Van Est Isomorphism(Theorem 4.6.3) and to settle some matters regarding measure homology andthe smearing procedure in Chapter 5.Let M be a smooth manifold throughout this section.

Definition 4.3.1. The space of smooth singular k-chains C∞k (M,R) is thefree R-vector space over the set of all smooth maps ∆k →M .We also define the space of smooth singular k-cochains Ck∞(M,R) as the(algebraic) dual of C∞k (M,R).

We obtain two complexes (C∞∗ (M,R), ∂∗), (C∗∞(M,R), d∗) with the usualboundary/coboundary maps.

Definition 4.3.2. The smooth singular homology spaces H∞∗ (M,R) are thehomology spaces of the complex (C∞∗ (M,R), ∂∗). Similarly we define thesmooth singular cohomology spaces H∗∞(M,R).By considering the subcomplex (C∗b,∞(M,R), d∗) of smooth singular k-cochainsthat are bounded on smooth singular simplices, we can also define the smoothbounded cohomology spaces H∗b,∞(M,R).

We have natural maps

ι∞ : H∞k (M,R)→ Hk(M,R) ,

since the complex (C∞∗ (M,R), ∂∗) is a subcomplex of the usual complexcomputing singular homology.Similarly, the restriction map Ck(M,R)→ Ck∞(M,R) induces

ι∞ : Hk(M,R)→ Hk∞(M,R)

andι∞b : Hk

b (M,R)→ Hkb,∞(M,R) .

The `1-norm on C∞k (M,R) induces a seminorm onH∞k (M,R) for each k ≥ 0.Moreover, the `∞-norm on Ckb,∞(M,R) induces a seminorm on Hk

b,∞(M,R)and the maps

Hkb,∞(M,R)→ Hk

∞(M,R)

induce seminorms also on each Hk∞(M,R).

Proposition 4.3.3. The maps ι∞, ι∞ and ι∞b are isometric isomorphismsfor each k ≥ 0 and each smooth manifold M .

76

Proof. It is possible to define a retraction Sk : Ck(M,R) → C∞k (M,R) sothat the composition

Ck(M,R)→ C∞k (M,R) → Ck(M,R)

is homotopic to the identity. The map Sk sends each singular simplex toa single smooth simplex; the homotopy can be chosen so as to send eachsimplex to a sum of at most k + 1 simplices. See Theorem 18.7 in [Lee12]for more details on this construction. The proposition easily follows.

Remark 4.3.4. Observe that the statement and proof of Theorem 4.2.9 stillwork (requiring no changes whatsoever) in the context of smooth singularcohomology.

Now consider the bilinear map

C∞p (M,R)× Ωp(M,R)→ R

(σ, ω) 7→∫σω

It allows us to define maps

Ip : Ωp(M,R)→ Cp∞(M,R) ,

that form a chain map thanks to Stokes’ Theorem.

Proposition 4.3.5. The induced maps Ip : HpDR(M,R) → Hp

∞(M,R) areisomorphisms for each p ≥ 0 and each smooth manifold M .

Proof. If M is diffeomorphic to an euclidean space, the Proposition is obvi-ous.In the general case, let Ukk∈N be a good open cover for Mn, i.e. an opencover having all finite intersections diffeomorphic to Rn. It is easy to prove,inductively on m, that the Proposition holds for each U1 ∪ ...∪Um; one onlyneeds to use the Mayer-Vietoris sequence (in both the involved cohomologi-cal theories) and the Five Lemma.Our assertion then follows immediately for all compact manifolds; the gen-eral case can be handled as in Lemma 5.6.5 in [AT11].

In smooth singular cohomology we can find a representative of the dualfundamental class related to the volume of simplices. In Chapter 5 willbecome clear why this is particularly convenient.

Corollary 4.3.6. Let Mn be a closed Riemannian manifold with volumeform volM . The smooth singular n-cocycle volM defined on smooth singularsimplices by

volM (σ) =

∫σvolM

represents vol(M) · ωM in Hn∞(M,R) ' Hn(M,R).

77

Proof. Let c =∑

α λα · σα be smooth singular chain representing [M ]; sinceM admits a smooth triangulation, we can assume that each σα is a simplexin a fixed smooth triangulation of M (up to permutation of the vertices).Then ωM (c) = 1 and volM (c) =

∑α λα · volM (σα) = vol(M).

The usual cup product in singular cohomology can be easily generalisedto give cup products

` : Cp∞(M,R)⊗ Cq∞(M,R)→ Cp+q∞ (M,R) ,

` : Hp∞(M,R)⊗Hq

∞(M,R)→ Hp+q∞ (M,R) ,

enjoying the same properties.We recall that the wedge product of forms on M defines a linear map

∧ : HpDR(M,R)⊗Hq

DR(M,R)→ Hp+qDR (M,R) .

The isomorphisms I∗ that we defined in this section are multiplicative withrespect to these products.

Theorem 4.3.7. The diagram

HpDR(M,R)⊗Hq

DR(M,R)

∧

Ip⊗Iq // Hp∞(M,R)⊗Hq

∞(M,R)

`

Hp+qDR (M,R)

Ip+q // Hp+q∞ (M,R)

commutes for each smooth manifold M and each p, q ∈ N.

Proof. This is Theorem 5.45 in [War83].

4.4 Other resolutions

Throughout this section, G will be a locally compact, Hausdorff, second-countable topological group, unless it is specified that we are assuming G tobe a Lie group. In addition, E will be a Banach G-module (even when weare simply considering continuous cohomology).The aim of this section is to provide additional complexes computing the con-tinuous cohomology and the continuous bounded cohomology of G (and of itsdiscrete subgroups) with coefficients in E; see Corollaries 4.4.3, 4.4.15, 4.4.20and 4.4.23.We could have adopted a more abstract approach and have shown that cer-tain resolutions of E are strong and relatively injective. Instead, we choseto construct explicit homotopy equivalences with the complexes associatedto the standard resolutions; this has the advantage of providing concreteisomorphisms between the cohomologies of the various complexes.

78

4.4.1 Continuous maps on coset spaces

Let H be a closed subgroup of G. Since E is a continuous Banach G-module,it is, in particular, a Fréchet G-module.The projections πk : Gk+1 → (G/H)k+1 induce maps

π∗ : C((G/H)k+1, E)→ C(Gk+1, E) ;

the left diagonal action of G on Gk+1 descends to an action on (G/H)k+1

and provides a structure of Fréchet G-module on C((G/H)k+1, E) (as inExample 1.5.15). It is clear that each π∗ is a G-map.The maps π∗ restrict to norm-non-increasing G-maps

π∗b : Cb((G/H)k+1, E)→ Cb(Gk+1, E) .

It is clear that π∗, π∗b are chain maps if we consider the customary coboundarymaps between the involved spaces.

Proposition 4.4.1. If H is a compact subgroup of G, there exists a contin-uous, G-equivariant chain map

τ :(C(G∗+1, E), d∗

)→(C((G/H)∗+1, E), d∗

)such that τ π∗ = id and such that π∗ τ is homotopic to the identity via acontinuous, G-equivariant homotopy.

Proof. Let µ be the Haar measure on H with µ(H) = 1. For f ∈ C(Gk+1, E),define τkf ∈ C((G/H)k+1, E) as

(τkf)(g0H, ..., gkH) :=

∫Hk+1

f(g0h0, ..., gkhk) dµk+1(h0, ..., hk) ,

where µk+1 is the product measure on Hk+1 induced by µ. Since µ is in-variant under left translations, this value does not depend on the chosenrepresentatives g0, ..., gk. It is easy to see that τkf is continuous by exploit-ing Theorem 1.5.19 and local continuous sections of G → G/H; it is alsoclear that τk is a G-map.Moreover, the maps τk form a chain map:

(τk+1dkf)(g0H, ..., gk+1H) =

=

∫Hk+2

dkf(g0h0, ..., gk+1hk+1) dµk+2(h0, ..., hk+1) =

=

k+1∑j=0

(−1)j ·∫Hk+2

f(g0h0, ..., gjhj , ..., gk+1hk+1) dµk+2(h0, ..., hk+1) =

=

k+1∑j=0

(−1)j ·∫Hk+1

f(g0h0, ..., gjhj , ..., gk+1hk+1) dµk+1(h0, ..., hj , ..., hk+1) =

79

=

k+1∑j=0

(−1)j · (τkf)(g0H, ..., gjH, ..., gk+1H) = (dkτkf)(g0H, ..., gk+1H) ,

where we have used the Fubini-Tonelli Theorem for the Bochner integral.The compositions τk π∗ are all the identity on C((G/H)k+1, E); instead,for each f ∈ C(Gk+1, E):

(π∗τkf)(g0, ..., gk) =

∫Hk+1

f(g0h0, ..., gkhk) dµk+1(h0, ..., hk) .

We will now construct G-maps T k : C(Gk+1, E) → C(Gk, E) realising a ho-motopy between the chain map π∗ τ and the identity.Define:

(T kf)(g0, ..., gk−1) :=

=k−1∑j=0

(−1)j ·∫Hk−j

f(g0, ..., gj , gjhj , ..., gk−1hk−1) dµk−j(hj , ..., hk−1) .

It is clear that T kf is continuous and that T k is a G-map. Now:

(dk−1T kf)(g0, ..., gk) =k∑j=0

(−1)j · (T kf)(g0, ..., gj , ..., gk) =

=

k∑j=0

(−1)j ·

[ ∑0≤i<j

(−1)i ·∫Hk−i

f(g0, ..., gi, gihi, ..., gjhj , ..., gkhk)·

· dµk−i(hi, ..., hj , ..., hk)+

+∑j<i≤k

(−1)i−1 ·∫Hk−i+1

f(g0, ..., gj , ..., gi, gihi, ..., gkhk) dµk−i+1(hi, ..., hk)

],

and(T k+1dkf)(g0, ..., gk) =

=

k∑i=0

(−1)i ·∫Hk−i+1

(dkf)(g0, ..., gi, gihi, ..., gkhk) dµk−i+1(hi, ..., hk) =

=

k∑i=0

(−1)i ·

[ ∑0≤j<i

(−1)j∫Hk−i+1

f(g0, ..., gj , ..., gi, gihi, ..., gkhk)·

· dµk−i+1(hi, ..., hk)+

80

+∑

i+2≤j≤k+1

(−1)j∫Hk−i

f(g0, ..., gi, gihi, ..., gj−1hj−1, ..., gkhk)·

· dµk−i(hi, ..., hj−1, ..., hk)

]+

+

k∑i=0

[∫Hk−i+1

f(g0, ..., gi−1, gihi, ..., gkhk) dµk−i+1(hi, ..., hk)+

−∫Hk−i

f(g0, ..., gi, gi+1hi+1, ..., gkhk) dµk−i(hi+1, ..., hk)

]=

= −(dk−1T kf)(g0, ..., gk) + (π∗τkf)(g0, ..., gk)− f(g0, ..., gk) .

Proposition 4.4.2. If H is a compact subgroup of G, there exists a contin-uous, norm-non-increasing, G-equivariant chain map

τb :(Cb(G∗+1, E), d∗

)→(Cb((G/H)∗+1, E), d∗

)such that τb π∗b = id and such that π∗b τb is homotopic to the identity viaa continuous, G-equivariant homotopy.

Proof. This proposition can be proven exactly like the previous one, definingτkb and T kb as the restrictions of τk and T k to bounded functions. It isevident that ‖τkb f‖∞ ≤ ‖f‖∞ and ‖T kb f‖∞ ≤ k · ‖f‖∞. So τkb and T kb mapbounded functions to bounded functions and are continuous with respect tothe Banach topologies on the spaces of continuous, bounded functions.

Corollary 4.4.3. If H is a compact subgroup of G, the continuous cohomol-ogy spaces Hk

c (G,E) can be computed via the complex:

0 C(G/H,E)G C((G/H)2, E)G ... .d0 d1

Moreover, the complex

0 Cb(G/H,E)G Cb((G/H)2, E)G ... .d0 d1

computes the continuous bounded cohomology spaces Hkcb(G,E) and the norm

‖ · ‖∞ on Cb((G/H)k+1, E)G induces the canonical seminorm.

Proof. For the first assertion, it suffices to observe that, since τ, π∗, T are G-equivariant, they descend to maps between the subcomplex of G-invariantfunctions, providing a homotopy equivalence with the standard complex.In the case of continuous bounded cohomology, since τb, π∗b are both norm-non-increasing, the norm induced on Hk

cb(G,E) by Cb((G/H)k+1, E)G mustcoincide with that induced by Cb(Gk+1, E)G, which is the canonical semi-norm.

81

4.4.2 Measurable maps on coset spaces

Definition 4.4.4. Let X be a topological space and consider the followingequivalence relation on Borel measures on X:

µ ∼ ν ⇔ µ and ν share the same null sets .

A measure class on a topological space X is an equivalence class for therelation ∼.

If µ ∼ ν are measures on X and µk, νk are the product measures on Xk,we have µk ∼ νk. Hence, a measure class on X defines measure classes onall powers Xk.

Example 4.4.5. On each topological space we can consider the measureclass associated to the counting measure. The only null set is the empty set.

Example 4.4.6. On a manifold M we can always consider the Lebesguemeasure class. It is the equivalence class of all the measures induced byvolume forms on M ; the associated null sets are precisely those Borel setswhose intersection with all charts is Lebesgue-null.On a Lie group, the Haar and Lebesgue measure classes coincide. Indeed,the Haar measure is induced by the Riemannian volume form associated toany G-invariant Riemannian metric on G.

Given a measure µ on the topological space X, we can consider the spaceL∞(X,µ) of Borel measurable, real-valued functions that are µ-essentially-bounded and define

L∞(X,µ) := L∞(X,µ)/V ,

where V is the subspace of functions that vanish µ-almost-everywhere. Thespace L∞(X,µ) is Banach with the norm

‖f‖∞ := inf M ∈ R | |f | ≤M µ-almost-everywhere .

Observe that the definition of L∞(X,µ) depends only on the measure classof µ; hence, when the considered measure class is clear from the context, wewill simply denote this space by L∞(X).Let R be a 1-dimensional Banach G-module. If G acts on the left on X andthis action preserves the measure class (i.e. sends null sets to null sets), wecan endow L∞(X) with the structure of Banach G-module given by

(g · f)(x) := g ·(f(g−1 · x)

);

we will denote the resulting G-module by L∞(X, R).Finally, we can define maps dk : L∞(Xk+1, µk+1)→ L∞(Xk+2, µk+2):

(dkf)(x0, ..., xk+1) =k+1∑j=0

(−1)j · f(x0, ..., xj , ..., xk+1) .

82

If we change f on a µk+1-null set, dkf changes on a µk+2-null set, so it iswell-defined; moreover, it is clear that dk is a G-map.

Proposition 4.4.7. Let M be a homogeneous Riemannian manifold en-dowed with the Lebesgue measure class. Let G be a Lie subgroup of I(M).The inclusion of complexes of Banach G-modules

i :(Cb(M∗+1, R), d∗

)→(L∞(M∗+1, R), d∗

)is a norm-non-increasing, G-equivariant chain map and there exists anothernorm-non-increasing, G-equivariant chain map

τ :(L∞(M∗+1, R), d∗

)→(Cb(M∗+1, R), d∗

)such that the compositions j τ and τ j are both homotopic to id via acontinuous, G-equivariant homotopy.

Proof. For each p ∈M let Bp be the closed ball around p with radius 1 andlet v be the volume of these balls; let volM be the volume form on M anddefine ω := (volM )/v.Define a map mk : L∞(Mk+1, R)→ Cb(Mk+1, R):

(mkf)(x0, ..., xk) :=

∫Bx0×...×Bxk

f(t0, ..., tk) · ωk+1(t0, ..., tk) .

The continuity of mkf follows from the Lebesgue Dominated ConvergenceTheorem and from the fact that all the balls Bx are compact, since M iscomplete. Moreover, ‖mk‖ ≤ 1.Since ω is G-invariant, we easily obtain that mk is G-equivariant:

mk(g·f)(x0, ..., xk) =

∫Bx0×...×Bxk

g·(f(g−1·t0, ..., g−1·tk)

)·ωk+1(t0, ..., tk) =

= g ·

(∫Bg−1·x0

×...×Bg−1·xk

f(t0, ..., tk) · ωk+1(t0, ..., tk)

)=

= g ·(

(mkf)(g−1 · x0, ..., g−1 · xk)

)=(g · (mkf)

)(x0, ..., xk) .

The maps mk form a chain map as one can easily see by imitating theproof of an analogous statement given for the maps τk during the proof ofProposition 4.4.1.We will now show that the compositions i m and m i are homotopicto the identity. We will construct maps T k : L∞(Mk+1, R) → L∞(Mk, R)satisfying i mk − id = dk−1T k + T k+1dk. Since T k will map continuousmaps to continuous maps, this will also prove that m i is homotopic to id.Define:

(T kf)(x0, ..., xk−1) :=

83

=

k−1∑j=0

(−1)j ·∫Bxj×...×Bxk−1

f(x0, ..., xj , tj , ..., tk−1) · ωk−j(tj , ..., tk−1) .

It is easy to check that the definition does not depend on the representativechosen for f in L∞(Mk+1). Checking that T indeed defines the requiredhomotopy can be handled just like in the last part of the proof of Proposi-tion 4.4.1.

Proposition 4.4.8. The previous proposition holds also if we consider themeasure class of the counting measure on M in the definition of the spacesL∞(Mk+1, R).

Proof. The proof of Proposition 4.4.7 carries over to this case without re-quiring any changes whatsoever.

Definition 4.4.9. Let µ be a left Haar measure on G and let G act triviallyon R. Then G is said to be amenable if there exists a functional

m : L∞(G,µ)→ R

with the following properties:

• ‖m‖ ≤ 1;

• m(χG) = 1;

• m(g · f) = m(f) for each g ∈ G and f ∈ L∞(G).

The functional m will be called a left-invariant mean.

Lemma 4.4.10. Compact groups are amenable.

Proof. Let µ be the Haar measure on G with µ(G) = 1. Define, for eachf ∈ L∞(G):

m(f) :=

∫Gf(x) dµ(x) .

All the required properties are evidently satisfied.

We will need the following technical lemma several times in the followingdiscussion.

Lemma 4.4.11. Let G be a Lie group and let H be a closed subgroup; con-sider the Lebesgue measure class on H. For p, q ≥ 0, fix a real-valued, Borelmeasurable, bounded function f : Gp+1 ×Gq → R and consider the functionF : Gp+1 ×Gq → L∞(Hp+1) given by:

F (x0, x1, ..., xp, y1, ..., yq)(h0, ..., hp) := f(x0h0, ..., xphp, y1, ..., yq) .

Then F is Borel measurable (considering in L∞(Hp+1) the Borel sets asso-ciated to its Banach topology).

84

Proof. We just have to show that, for each open subset U of L∞(Hp+1), thepreimage F−1(U) is Borel measurable.Without loss of generality, U is of the form

U = h ∈ L∞(Hp+1) | ‖h− g‖∞ < ε ,

for some g ∈ L∞(Hp+1) and ε > 0. Define Ω ⊆ Gp+1 ×Gq ×Hp+1:

Ω := (x, y, h) | |f(x0h0, ..., xphp, y1, ..., yq)− g(h0, ..., hp)| ≥ ε .

Clearly, F−1(U) is precisely the set

(x, y) | Ωx,y is null in Hp+1 ⊆ Gp+1 ×Gq ,

whereΩx,y := (h0, ..., hp) ∈ Hp+1 | (x, y, h0, ..., hp) ∈ Ω .

Since Ω is clearly a Borel set, the Fubini-Tonelli Theorem guarantees thatso is F−1(U).

Remark 4.4.12. Products of amenable groups are amenable.We are particularly interested in the construction of a left-invariant meanmk on the group Hk+1 when H is amenable and a mean m on H is given.We will proceed by induction, starting with m0 := m.Suppose that the left-invariant mean mk−1 has been defined. Define, for abounded, Borel measurable function f : Hk+1 → R:

fh1,...,hk(h) := f(h, h1, ..., hk)

and Ff : Hk → L∞(H):

Ff (h1, ..., hk) := fh1,...,hk .

The map Ff is Borel measurable thanks to Lemma 4.4.11; hence, also mFfis Borel measurable and defines an element of L∞(Hk). If we modify f on anull set, the map m Ff changes only on a null set. Now define:

mk(f) := mk−1 (m Ff ) .

It is easy to check that mk is well-defined and 1-Lipschitz. Moreover:

mk(χHk+1) = mk−1(m Fχ

Hk+1

)= mk−1(χHk) = 1 ,

and, for x = (x0, ..., xk) ∈ Hk+1,

Fx·f (h1, ..., hk) = (x · f)h1,...,hk = x0 · (fx−11 h1,...,x

−1k hk

) ,

so that

mk(x · f) = mk−1(m Fx·f

)= mk−1 ((x1, ..., xk) · (m Ff )) =

= mk−1 (m Ff ) = mk(f) .

85

Proposition 4.4.13. Let G be a Lie group a let A be a closed, amenablesubgroup; let R be a 1-dimensional Banach G-module. Consider the Lebesguemeasure class on both G and G/A.The chain map induced by the projection π : G→ G/A

π∗ : L∞((G/A)∗+1, R) → L∞(G∗+1, R)

is G-equivariant and norm-non-increasing. Moreover, there exists an equiv-ariant, norm-non-increasing chain map

τ : L∞(G∗+1, R)→ L∞((G/A)∗+1, R)

such that τ π∗ is the identity and π∗ τ is homotopic to the identity via acontinuous, G-equivariant homotopy.

Proof. Let m be a left-invariant mean on A; recall that in Remark 4.4.12 wehave defined left-invariant means mk on Ak+1 for each k ≥ 0.Define the maps τk : L∞(Gk+1, R)→ L∞((G/A)k+1, R):

(τkf)(g0A, ..., gkA) := mk ((a0, ..., ak) 7→ f(g0a0, ..., gkak)) .

It is clear that this value depends neither on the representative chosen forf in L∞(Gk+1), nor on the representatives g0, ..., gk for the cosets of A.Moreover, τkf is Borel measurable thanks to Lemma 4.4.11 and to the factthat mk is continuous. We clearly have ‖τk‖ ≤ 1 and τk is G-equivariant.Observe that the maps τk form a chain map:

(dkτkf)(g0A, ..., gk+1A) =k+1∑j=0

(−1)j · (τkf)(g0A, ..., gjA, ..., gk+1A) =

=k+1∑j=0

(−1)j ·mk ((a0, ..., aj , ..., ak+1) 7→ f(g0a0, ..., gjaj , ..., gk+1ak+1)) =

=k+1∑j=0

(−1)j ·mk+1 ((a0, ..., ak+1) 7→ f(g0a0, ..., gjaj , ..., gk+1ak+1)) =

= mk+1(

(a0, ..., ak+1) 7→ dkf(g0a0, ..., gk+1ak+1))

=

= (τk+1dkf)(g0A, ..., gk+1A) .

Moreover, all the compositions τk π∗ are evidently the identity, whereas:

(π∗τkf)(g0, ..., gk) = mk ((a0, ..., ak) 7→ f(g0a0, ..., gkak)) .

Define maps T k : L∞(Gk+1, R)→ L∞(Gk, R):

(T kf)(g0, ..., gk−1) :=

86

=

k−1∑j=0

(−1)j ·mk−j ((aj , ..., ak) 7→ f(g0, ..., gj , gjaj , ..., gkak)) .

The map T kf is well defined and it is measurable due to Lemma 4.4.11.Moreover, it is clear that T k is G-equivariant and satisfies ‖T k‖ ≤ k. Finally,one can check that the maps T k define a homotopy between π∗ τ and theidentity; it suffices to reproduce the computations at the end of the proof ofProposition 4.4.1.

Proposition 4.4.14. The previous proposition holds also if we consider themeasure class of the counting measure on G and G/A in the definition of thespaces L∞(Gk+1, R) and L∞((G/A)k+1, R).

Proof. Again, it suffices to reproduce exactly the proof of Proposition 4.4.13.We remark that even in this case, we have to keep considering the Lebesguemeasure class on A, since the left-invariant means are defined in this context.

Corollary 4.4.15. Let G be a Lie group and A a closed, amenable subgroup.Then the complex

0 L∞(G/A, R)G L∞((G/A)2, R)G ...d0 d1

computes the continuous bounded cohomology H∗cb(G, R) and the norm ‖ · ‖∞on L∞((G/A)k+1, R) induces the canonical seminorm on Hk

cb(G, R).This is true both if we consider the Lebesgue measure class on G,G/A andif we consider the measure class of the counting measure on G,G/A.

Proof. Fix a left-invariant Riemannian metric G, so that it becomes a ho-mogeneous manifold with G embedded in I(G) by left-translations. Propo-sitions 4.4.7 and 4.4.8 yield a homotopy equivalence between the complexes(

L∞(G∗+1, R)G, d∗)

and(Cb(G∗+1, R)G, d∗

),

whereas Propositions 4.4.13 and 4.4.14 provide a homotopy equivalence be-tween the complexes(

L∞((G/A)∗+1, R)G, d∗)

and(L∞(G∗+1, R)G, d∗

).

Since all the maps involved in these homotopy equivalences are norm-non-increasing, the corollary follows immediately.

We obtain the following generalisation of Lemma 4.4.10.

Corollary 4.4.16. If G is an amenable Lie group, then Hkcb(G,R) = 0

for each k ≥ 1.

87

Proof. Choosing G = A in Corollary 4.4.15, each L∞((G/A)k, R)G is one-dimensional and the coboundary maps are alternately zero and isomor-phisms.

All the results of this subsection hold also with coefficients in a generalBanach G-module; we did not carry out the proofs in this more generalcontext to avoid technicalities regarding the measurability of vector-valuedfunctions.

4.4.3 Discrete subgroups

Let now Γ be a discrete subgroup of G. Since Γ acts on G by left translations,we can view each C(Gk+1, E) as a Fréchet Γ-module and each Cb(Gk+1, E)as a Banach Γ-module.

Definition 4.4.17. A Bruhat function for the action of Γ on G is a contin-uous function h : G→ [0,+∞) satisfying

•∑

γ∈Γ h(γ−1 · g0) = 1 for each g0 ∈ G;

• (Γ ·K) ∩ (Supp h) is compact for each compact subset K ⊆ G.

Lemma 4.4.18. We can always find a Bruhat function h with h(e) = 1.

Proof. The projection π : G→ G/Γ is a covering map and the space G/Γ ismetrisable by the Nagata-Smirnov Theorem; hence G/Γ is paracompact. LetUα be a locally finite open cover of G/Γ such that each Uα has compactclosure contained in an evenly covered open subset. Choose a partition ofunity ρα subordinate to Uα and choose for each α a connected compo-nent Uα of π−1(Uα). Define the continuous functions ηα : G→ [0, 1] as ρα πon Uα and 0 outside Uα.The function

h :=∑α

ηα

is well-defined on G, continuous and it takes values in [0, 1].Observe that, for each compact subset K ⊆ G, the subset (Γ ·K)∩ (Supp h)is compact. Otherwise, we would be able to find infinitely many distinctelements γn of Γ such that

(γn ·K) ∩ (Supp h) 6= ∅ ,

and the γn ·K would intersect only finitely many Uα since they all project tothe same compact subset π(K) ⊆ G/Γ and the cover Uα is locally finite.Hence there would be some α0 such that

(γn ·K) ∩ Uα0 6= ∅

88

holds for infinitely many n ∈ N and this violates the discreteness of Γ, sincethe closure of Uα0 is compact.The requirement ∑

γ∈Γ

h(γ−1 · g0) = 1

is clearly satisfied for each g0 ∈ G and also the requirement h(e) = 1 can beeasily achieved by an appropriate choice of the partition ρα.

Proposition 4.4.19. Let Γ be a discrete subgroup of G. The chain mapinduced by the inclusion i : Γ→ G:

i∗ : C(G∗+1, E)→ C(Γ∗+1, i∗E)

is Γ-equivariant and continuous. Moreover, we can find a continuous, Γ-equivariant chain map

τ : C(Γ∗+1, i∗E)→ C(G∗+1, E)

so that i∗ τ = id and τ i∗ is homotopic to the identity via a continuous,Γ-equivariant homotopy.An analogous statement holds for

i∗ : Cb(G∗+1, E)→ Cb(Γ∗+1, i∗E) ,

where, in addition, the homotopy inverse τb can be chosen to be norm-non-increasing.

Proof. Let h be a Bruhat function for the action of Γ on G and assume thath(e) = 1. Define a map τk : C(Γk+1, i∗E)→ C(Gk+1, E):

(τkf)(g0, ..., gk) :=∑

γ0,...,γk∈Γ

h(γ−10 g0) · ... · h(γ−1

k gk) · f(γ0, ..., γk) .

The map τk is clearly well-defined and continuous; moreover, it sends boundedfunctions to bounded functions without increasing their norm. It is easyto check that the maps τk provide a chain map; furthermore, they are Γ-equivariant:

(γ · τkf)(g0, ..., gk) = γ ·(

(τkf)(γ−1g0, ..., γ−1gk)

)=

= γ ·

∑γ0,...,γk∈Γ

h(γ−10 γ−1g0) · ...h(γ−1

k γ−1gk) · f(γ0, ..., γk)

=

=∑

γ0,...,γk∈Γ

h(γ−10 g0) · ...h(γ−1

k gk) ·(γ ·(f(γ−1γ0, ..., γ

−1γk)) )

=

89

= (τk(γ · f))(g0, ..., gk) .

Finally, i∗ τ = id since h(e) = 1 implies that h vanishes on Γ \ 0.Moreover, the composition τ i∗ is homotopic to the identity via the mapsT k : C(Gk+1, E)→ C(Gk, E):

(T kf)(g0, ..., gk−1) =

=k−1∑j=0

(−1)j ·

∑γj ,...,γk∈G

h(γ−1j gj) · ...h(γ−1

k gk) · f(g0, ..., gj , γj , ..., γk)

.

This can be checked just as in the proof of Proposition 4.4.1. Moreover, it isevident that T k sends bounded maps to bounded maps and ‖T k‖ ≤ k.

Corollary 4.4.20. Let Γ be a discrete subgroup of G. Then the cohomologyspaces H∗(Γ, i∗E) can be computed via the complex:

0 C(G,E)Γ C(G2, E)Γ C(G3, E)Γ ... .d0 d1 d2

Moreover, the bounded cohomology spaces H∗b (Γ, E) can be computed via thecomplex:

0 Cb(G,E)Γ Cb(G2, E)Γ Cb(G3, E)Γ ... ,d0 d1 d2

which induces the canonical seminorm on H∗b (Γ, E).

4.4.4 Alternating maps

Let X be a locally compact, Hausdorff, second-countable topological space.

Definition 4.4.21. A function f : Xk+1 → E is said to be alternating if,for each permutation σ ∈ Sk+1, we have

f(xσ(0), ..., sσ(k)) = εσ · f(x0, ..., xk), ∀x0, ..., xk ∈ X,

where εσ denotes the sign of σ.

We can consider the Fréchet space Calt(Xk+1, E) of continuous, alter-nating functions and the Banach space of bounded, continuous, alternatingfunctions Cb,alt(Xk+1, E). If G acts on X, these spaces inherit structuresof G-module from C(Xk+1, E) and Cb(Xk+1, E); indeed, they are closed, G-invariant subspaces.Moreover, fixing a measure class on X, we can consider the closed subspaceL∞alt(X

k+1) ⊆ L∞(Xk+1) of functions having an alternating representative.

90

When G acts on X, this space also inherits a structure of Banach G-module.For a function f : Xk+1 → E, we can define

(Alt f)(x0, ..., xk) :=1

(k + 1)!

∑σ∈Sk+1

εσ · f(xσ(0), ..., xσ(k)) ,

which is an alternating function Xk+1 → E.It is clear that this allows us to define continuous retractions on the closedsubspaces of alternating functions:

Altk : C(Xk+1, E)→ Calt(Xk+1, E) ,

Altk : Cb(Xk+1, E)→ Cb,alt(Xk+1, E) ,

Altk : L∞(Xk+1)→ L∞alt(Xk+1) ,

and, in the second and third case, we have ‖Altk‖ ≤ 1. If G acts on X, themaps Altk are G-equivariant with respect to the induced actions.

Proposition 4.4.22. In all three cases Alt is a chain map homotopic to theidentity via a G-equivariant, continuous homotopy.

Proof. First, we check that Alt is a chain map. For each f : Xk+1 → E, wehave:

(dkAltkf)(x0, ..., xk+1) =

k+1∑j=0

(−1)j · (Altkf)(x0, ..., xj , ..., xk+1) =

=k+1∑j=0

(−1)j

(k + 1)!·∑

σ∈Sk+1

εσ · f(xσ(0), ..., xj , ..., xσ(k+1)) ,

(Altk+1dkf)(x0, ..., xk+1) =1

(k + 2)!·∑

σ∈Sk+2

εσ · dkf(xσ(0), ..., xσ(k+1)) =

=1

(k + 2)!·∑

σ∈Sk+2

k+1∑j=0

(−1)j · εσ · f(xσ(0), ..., xσ(j), ..., xσ(k+1)) .

Consider a summand in (dkAltkf)(x0, ..., xk+1):

(−1)jεσ(k + 1)!

· f(xσ(0), ..., xj , ..., xσ(k+1)) ;

for each i = 0, ..., k + 1 there exists exactly one τi ∈ Sk+2 such that

(xσ(0), ..., xj , ..., xσ(k+1)) = (xτ(0), ..., xτ(i), ..., xτ(k+1))

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and it is easy to check that (−1)iετ = (−1)jεσ. Hence, the sum of the k + 2terms in (Altk+1dkf) corresponding to the τi equals

(−1)jεσ(k + 1)!

· f(xσ(0), ..., xj , ..., xσ(k+1)) ;

this shows that

(dkAltkf)(x0, ..., xk+1) = (Altk+1dkf)(x0, ..., xk+1) .

Now we proceed to show that Alt is homotopic to the identity. Define, forf : Xk+1 → E:

(T kf)(x0, ..., xk−1) :=

k−1∑j=0

(−1)j

(k − j)!·∑

σ∈Sk−j

f(x0, ..., xj , xσ(j), ..., xσ(k−1)) .

Continuity and equivariance of T are evident; we will now show that therelation dk−1T k + T k+1dk = Altk − id holds. Observe that:

(dk−1T kf)(x0, ..., xk) =

k∑i=0

(−1)i · (T kf) (x0, ..., xi, ..., xk) =

=

k∑i=0

[i−1∑j=0

(−1)i+j

(k − j)!·∑

σ∈Sk−j

εσ · f(x0, ..., xj , xσ(j), ..., xi, ..., xσ(k)

)+

+

k∑j=i+1

(−1)i+j−1

(k − j + 1)!·

∑σ∈Sk−j+1

εσ · f(x0, ..., xi, ..., xj , xσ(j), ..., xσ(k)

) ],

(T k+1dkf)(x0, ..., xk) =

=

k∑j=0

(−1)j

(k − j + 1)!·

∑σ∈Sk−j+1

εσ · (dkf)(x0, ..., xj , xσ(j), ..., xσ(k)

)=

=

k∑j=0

∑σ∈Sk−j+1

(−1)jεσ(k − j + 1)!

·

[j−1∑i=0

(−1)i ·f(x0, ..., xi, ..., xj , xσ(j), ..., xσ(k)

)+

+

k∑i=j

(−1)i+1 · f(x0, ..., xj , xσ(j), ..., xσ(i), ..., xσ(k)

)+

+ (−1)j · f(x0, ..., xj−1, xσ(j), ..., xσ(k)

) ].

It is easy to see that the second double summation in (dk−1T kf)(x0, ..., xk)cancels with the first triple summation in (T k+1dkf)(x0, ..., xk). Moreover,

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an argument similar to that used to prove that Alt is a chain map showsthat the first double summation in (dk−1T kf)(x0, ..., xk) cancels with allthe summands in the second triple summation in (T k+1dkf)(x0, ..., xk) thatcorrespond to permutations not taking i to j. In conclusion:

(dk−1T kf + T k+1dkf)(x0, ..., xk) =

=∑

0≤j≤i≤k

∑σ∈Sk−j+1

σ(i)=j

(−1)i+j+1εσ(k − j + 1)!

· f(x0, ..., xj , xσ(j), ..., xσ(i), ..., xσ(k)

)+

+

k∑j=0

∑σ∈Sk−j+1

εσ(k − j + 1)!

· f(x0, ..., xj−1, xσ(j), ..., xσ(k)

).

Carrying out the summation over i, we obtain by the usual argument:∑0≤j≤i≤k

∑σ∈Sk−j+1

σ(i)=j

(−1)i+j+1εσ(k − j + 1)!

· f(x0, ..., xj , xσ(j), ..., xσ(i), ..., xσ(k)

)=

= −k∑j=0

∑σ∈Sk−j

εσ(k − j)!

· f(x0, ..., xj , xσ(j+1), ..., xσ(k)

).

Finally:(dk−1T kf + T k+1dkf)(x0, ..., xk) =

= −k∑j=0

∑σ∈Sk−j

εσ(k − j)!

· f(x0, ..., xj , xσ(j+1), ..., xσ(k)

)+

+k∑j=0

∑σ∈Sk−j+1

εσ(k − j + 1)!

· f(x0, ..., xj−1, xσ(j), ..., xσ(k)

)=

= (Altkf)(x0, ...xk)− f(x0, ..., xk) .

Corollary 4.4.23. Let G be a Lie group, H a compact subgroup and A aclosed, amenable subgroup. Let E be a continuous Banach G-module and Ra 1-dimensional Banach G-module.The continuous cohomology spaces H∗c (G,E) can be computed via the reso-lution

0 Calt(G/H,E)G Calt((G/H)2, E)G ... .d0 d1

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Moreover, the resolutions

0 Cb,alt(G/H,E)G Cb,alt((G/H)2, E)G ... .d0 d1

0 L∞alt(G/A, R)G L∞alt((G/A)2, R)G ... .d0 d1

compute the spaces H∗cb(G,E) and H∗cb(G, R), respectively, and induce thecanonical seminorm on them. On G/A we can consider both the measureclass of the counting measure and the Lebesgue measure class.

4.5 Lattices in Lie groups

This section is devoted to the study of the connections between the contin-uous and continous bounded cohomology of a group and that of its discretesubgroups (under suitable hypotheses). We will start by introducing thenotion of lattice and carry on obtaining probably the deepest results in thisdissertation, namely Theorem 4.5.3 and Corollary 4.5.6. They will prove tobe invaluable tools in the next chapters. We will build on the relationshipobtained in Theorem 4.2.9 between the cohomology of a locally symmetricspace of noncompact type and that of its fundamental group, binding boththese concepts to the cohomology of the isometry group of the universal cov-ering.We start by considering a locally compact, Hausdorff topological group Gwith a left Haar measure µ.If Γ < G is a discrete subgroup, the covering map p : G→ Γ\G allows us todefine a measure µ on Γ\G which is induced by µ. If U is an evenly coveredopen subset of Γ\G and U0 is a connected component of p−1(U), we candefine µ on the Borel subsets of U by pushing forward the restriction of µto the Borel subsets of U0 via p. Clearly, this definition does not dependon the chosen connected component U0 and, if V is another evenly coveredopen subset of Γ\G, the measures defined on U and V coincide on U ∩ V .Hence it is possible to patch together all these measures to obtain a Borelmeasure µ on Γ\G.

Definition 4.5.1. Let G be a locally compact, Hausdorff topological groupand µ a left Haar measure. A discrete subgroup Γ < G is said to be a latticeif the measure µ induced by µ on Γ\G is finite.A lattice Γ is called a cocompact lattice if Γ\G is compact.

Note that the homeomorphism of G given by taking the inverse of eachelement induces a homeomorphism between the coset spaces G/Γ and Γ\G.

Proposition 4.5.2. A locally compact, Hausdorff topological group G con-taining a lattice Γ must be unimodular.

94

Proof. Let µ be a left-invariant Haar measure on G and µ the finite measurethat it induces on the quotient Γ\G.Each right translation in G defines a homeomorphism Rg : Γ\G → Γ\G; inparticular, (

(Rg)∗µ)

(Γ\G) = µ(Γ\G) .

However, (Rg)∗µ is again a left-invariant Haar measure on G, since left andright translations commute. Hence, by the uniqueness of Haar measures,there exists a positive real number λ with (Rg)∗µ = λ ·µ. As a consequence,(Rg)∗µ = λ · µ holds as well. But (Rg)∗µ and µ have the same total mea-sure, so it must be λ = 1. This means that µ is invariant under all righttranslations.

The following result provides an essential connection between the coho-mology of a group and that of its lattices. It is a key-point for the results ofChapters 5 and 6.

Theorem 4.5.3. Let G be a locally compact, Hausdorff, second-countabletopological group and Γ a lattice in G with inclusion map i : Γ → G. Let Ebe a continuous Banach G-module.

1. The morphismi∗ : Hk

cb(G,E)→ Hkb (Γ, i∗E)

is an isometric embedding for each k ≥ 0.

2. If Γ is a cocompact lattice, then the morphism

i∗ : Hkc (G,E)→ Hk(Γ, i∗E)

is injective for each k ≥ 0 and it preserves the seminorms.

Proof. 1. We will construct 1-Lipschitz transfer morphisms

τb : Hkb (Γ, i∗E)→ Hk

cb(G,E)

satisfying τb i∗ = id. This will be enough to prove the first part ofthe theorem, since i∗ is seminorm-non-increasing.The map

i∗ : Hkcb(G,E)→ Hk

b (Γ, i∗E)

is induced by the chain map:

i∗ : Cb(G∗+1, E)G → Cb(Γ∗+1, i∗E)Γ,

which we can factor as

Cb(G∗+1, E)G → Cb(G∗+1, i∗E)Γ → Cb(Γ∗+1, i∗E)Γ .

95

Hence, by Corollary 4.4.20, it suffices to construct a norm-non-increasingleft inverse for the inclusion

Cb(G∗+1, E)G → Cb(G∗+1, i∗E)Γ .

Let µ be a left-invariant Haar measure on G and µ the finite measureinduced on Γ\G. Define a map τkb : Cb(Gk+1, i∗E)Γ → Cb(Gk+1, E)G

by:

(τkb f)(g0, ..., gk) :=1

µ(Γ\G)

∫Γ\G

x−1 · f(x · g0, ..., x · gk) dµ(x).

Observe that the function that we are integrating is well-defined onΓ\G (thanks to the Γ-invariance of f); moreover, it is continuous andbounded. So τkb f is continuous by Proposition 1.5.19 and in addition‖τkb f‖∞ ≤ ‖f‖∞, i.e. ‖τkb ‖ ≤ 1.Observe that τkb takes values in the G-invariant maps; indeed, since µis right-invariant due to Proposition 4.5.2:

τkb (f)(gg0, ..., ggk) =1

µ(Γ\G)

∫Γ\G

x−1 · f(xg · g0, ..., xg · gk) dµ(x) =

=1

µ(Γ\G)

∫Γ\G

(xg−1)−1 · f(x · g0, ..., x · gk) dµ(x) =

= g ·

(1

µ(Γ\G)

∫Γ\G

x−1 · f(x · g0, ..., x · gk) dµ(x)

)=

g ·(τkb (f)(g0, ..., gk)

).

Finally, it is obvious that τkb i∗ = id and that τb is a norm-non-increasing chain map.

2. Again, thanks to Corollary 4.4.20, it suffices to construct a norm-non-increasing left inverse for the inclusion

C(G∗+1, E)G → C(G∗+1, i∗E)Γ.

Maps τk : C(Gk+1, i∗E)Γ → C(Gk+1, E)G can be defined by the sameformula that we used for τkb . In this case, they are well-defined (even iff may not be bounded) because Γ\G is compact and we are integratinga continuous function of x. We obtain a chain map

τ∗ : C(G∗+1, i∗E)Γ → C(G∗+1, E)G

and morphismsτ : Hk(Γ, i∗E)→ Hk

c (G,E)

96

satisfying τ i∗ = id. This proves that i∗ : Hkc (G,E) → Hk(Γ, E) is

injective for each k ≥ 0.To see that also this immersion is isometric, simply observe that thefollowing diagram commutes for each k

Hk(G,E) i∗ // Hk(Γ, E)

τ // Hk(G,E)

Hkb (G,E)

ιk

OO

i∗ // Hkb (Γ, E)

ιk

OO

τb // Hkb (G,E) ,

ιk

OO

as at the level of cochains it is given by

C(Gk+1, E)G i∗ // C(Gk+1, E)Γ τk // C(Gk+1, E)G

Cb(Gk+1, E)G?

OO

i∗ // Cb(Gk+1, E)Γ?

OO

τkb // Cb(Gk+1, E)G .?

OO

The map i∗ : Hkc (G,E) → Hk(Γ, E) is seminorm-non-increasing, since

the map i∗ : Hkbc(G,E) → Hk

b (Γ, E) is. If there were α ∈ Hkc (G,E)

with ‖i∗(α)‖ < ‖α‖, we could find some β ∈ Hkb (Γ, E) with ιk(β) =

i∗(α) and ‖β‖ < ‖α‖. But then:

ιkτb(β) = τιk(β) = τi∗(α) = α

and‖α‖ = ‖ιkτb(β)‖ ≤ ‖τb(β)‖ ≤ ‖β‖ < ‖α‖ ,

which is absurd. We conclude that i∗ is an isometric embedding alsoamong the continuous cohomology spaces.

We now proceed to obtain a more concrete version of Theorem 4.5.3,relating the cohomology of a Riemannian manifold with that of the isome-try group of its universal covering (under suitable hypotheses) (see Corol-lary 4.5.6).Let M be a connected Riemannian manifold with homogeneous universalcovering M and consider the group Γ < I(M) of covering automorphismsof M →M . It is an easy consequence of Theorem 1.4.2 that M is compactif and only if Γ\I(M) is. In particular, thanks to Proposition 4.5.2 (and, inthis case, even without the hypothesis of homogeneity):

Corollary 4.5.4. Let M be a Riemannian manifold admitting a locally iso-metric covering map onto a compact manifold. Then I(M) is unimodular.

The next lemma shows that, assuming that M is homogeneous, we canalso conclude that Γ is a lattice if and only if M has finite volume.

97

Lemma 4.5.5. 1. Let π : E → B be a smooth fibre bundle with fibre F ;call Fx the fibre above x ∈ B. Suppose that we are given a Riemannianmetric on B and a Riemannian metric along the fibres of E. Then wecan define a measure µ on the Borel sets of E as:

µ(A) :=

∫BvolFx(A ∩ Fx) dvolB(x).

2. Let M be a homogeneous Riemannian manifold with isometry group G;fix some x0 ∈ M and let H be the stabiliser of x0 in G.Then, the map πx0 : G → M ' G/H defined in Proposition 2.1.4 is afibre bundle with fibre H. The fibres of πx0 are precisely the left cosetsof H; identifying them with H, each left-invariant metric on H withvolume 1 defines a metric along the fibres of G → M . The measurethat we obtain on G (proceeding as in part 1) is a left-invariant Haarmeasure on G.

3. Suppose, in addition, that M is the universal covering of a Rieman-nian manifold M and let Γ < G be the group of automorphisms of thecovering map. Then Γ is a lattice in G if and only if M has finitevolume.

Proof. 1. We only have to check that the value µ(A) is well-defined, sinceadditivity is obvious.First, it makes sense to consider volFx(A ∩ Fx) since A ∩ Fx is a Borelset in Fx. Moreover, the function x 7→ volFx(A∩Fx) is a Borel functionon B; indeed, it is enough to check it for the restriction to an open setwhere π is of the form Rn×Rm → Rn and in this case it is simply theFubini-Tonelli Theorem.

2. The map πx0 is a proper submersion and, hence, a fibre bundle.To see that the measure µ defined as in point 1 is a Haar measure, weappeal to the Haar Uniqueness Theorem. It is clear that µ is finite oncompact sets and positive on open ones, so we only have to check thatit is left-invariant. This is a consequence of the fact that the action ofG on M preserves the volume form and of the fact that the maps

gH → g′gH

gh→ g′gh

preserve the metric on the fibres; the second statement follows fromthe fact that we have chosen a metric on H that is invariant under lefttranslations by elements of H and we have defined the metric on gHby pushing it forward via Lg.

98

3. By the previous description, there is a left-invariant Haar measure µ1

on G satisfyingµ1

(π−1x0 (A)

)= vol

M(A)

for each Borel subset A ⊆ M . We get an induced measure µ1 on thequotient space Γ\G.By choosing a Borel subset ∆ ⊆ M projecting bijectively to M , we seethat π−1

x0 (∆) is a measurable subset in Γ projecting bijectively to Γ\Gand

µ1(Γ\G) = µ1

(π−1x0 (∆)

)= vol

M(∆) = volM (M).

Corollary 4.5.6. Let M be a connected, aspherical Riemannian manifoldwith universal covering M . Let R be the vector space R endowed with theaction of G = I(M) that consists of the multiplication by ±1 depending onthe considered isometry being orientation-preserving or not. Then:

1. if M is finite-volume and M is homogeneous, we have

Hkcb(G, R) → Hk

b (M,R)

isometrically for each k ≥ 0;

2. if M is compact, we also have

Hkc (G, R) → Hk(M,R)

isometrically for each k ≥ 0.

Proof. Let Γ be the automorphism group of the covering map M → M .Since M is aspherical and simply connected, it is contractible by Whitehead’sTheorem (all smooth manifolds admit triangulations and hence CW-complexstructures); hence Theorem 4.2.9 and Remark 4.2.10 provide isometric iso-morphisms H∗b (M,R) ' H∗b (Γ,R), H∗(M,R) ' H∗(Γ,R).If M is finite-volume and M is homogeneous, Γ is a lattice in G because ofLemma 4.5.5 and Theorem 4.5.3 proves the first statement.If M is compact, Γ\G is as well by the Ascoli-Arzelà Theorem and, again,Theorem 4.5.3 proves our assertion.

Remark 4.5.7. The previous corollary applies, in particular, to all locallysymmetric spaces of noncompact type, or, more generally, to all nonpositivelycurved locally symmetric (or even locally homogeneous) spaces.

99

4.6 Cup product

Let G be a locally compact, Hausdorff, second-countable topological group.

Definition 4.6.1. Let X be a set. We define the cup product of functionsφ : Xk+1 → R and ψ : Xh+1 → R as the function φ ` ψ : Xk+h+1 → R:

(φ ` ψ)(x0, ..., xk+h) := φ(x0, ..., xk) · ψ(xk, ..., xk+h), ∀x0, ..., xk+h ∈ X.

Defining Ck(X,R) as the set of functions Xk+1 → R and considering theusual maps

dk : Ck(X,R)→ Ck+1(X,R) ,

the following relation is easy to check

dk+h(φ ` ψ) = (dkφ) ` ψ + (−1)k · φ ` (dhψ) .

If X is endowed with a topology, cup product of continuous maps is againcontinuous. Hence, we obtain well-defined maps in cohomology

` : Hkc (G,R)⊗Hh

c (G,R)→ Hk+hc (G,R) .

Since ‖φ ` ψ‖∞ ≤ ‖φ‖∞ · ‖ψ‖∞ whenever φ, ψ are bounded, we also get

` : Hkcb(G,R)⊗Hh

cb(G,R)→ Hk+hcb (G,R) .

Furthermore, ifι∗ : H∗cb(G,R)→ H∗c (G,R)

is the comparison morphism, we clearly have

ιk+h(α ` β) = ιk(α) ` ιh(β),

for all bounded classes α ∈ Hkcb(G,R), β ∈ Hh

cb(G,R).For a continuous morphism of groups f : H → G, it is also easy to checkthat:

f∗(α ` β) = f∗(α) ` f∗(β) .

Observe that, given a Borel measure on X, the cup product

` : L∞(Xk+1,R)⊗ L∞(Xh+1,R)→ L∞(Xk+h+1,R),

is well-defined too. Indeed, it is easy to check that f ` g does not dependon the representatives in L∞ that have been chosen for f, g.

Remark 4.6.2. Consider cohomology classes α ∈ Hkcb(G,R), β ∈ Hh

cb(G,R)and let A be a closed, amenable subgroup of G. We can choose repre-sentatives f ∈ L∞((G/A)k+1,R)G and g ∈ L∞((G/A)h+1,R)G (see Corol-lary 4.4.15). Then, the function f ` g ∈ L∞((G/A)k+h+1,R)G represents

100

the cohomology class α ` β.Indeed, if τ∗ is the chain map defined in Proposition 4.4.13, it is easy toprove by induction on k that τk+h(f ` g) = τk(f) ` τh(g). Hence, itsuffices to prove that τk(f) ` τh(g) represents α ` β. But, since τk(f)and τh(g) represent α and β respectively, we can find φ ∈ L∞(Gk,R)G andψ ∈ L∞(Gh,R)G such that

τk(f) + dk−1φ ∈ Cb(Gk+1,R)G and τh(g) + dh−1ψ ∈ Cb(Gh+1,R)G .

Then α ` β is represented by:

(τk(f) + dk−1φ) ` (τh(g) + dh−1ψ) =

= (τk(f) ` τh(g)) + dk+h−1(φ ` (τh(g) + dh−1ψ) + (−1)k · (τk(f) ` ψ)

).

Since the inclusion Cb(G∗+1,R)G → L∞(G∗+1,R)G corresponds to the iden-tity in cohomology (Proposition 4.4.13), we conclude that τk(f) ` τh(g)represents α ` β.

We recall that in Section 3.4 we obtained the Van Est Isomorphism,i.e. an identification between the continuous cohomology of a connected LiegroupG and the space ofG-invariant differential forms on a contractible cosetspace M associated to a connected, compact subgroup (Theorem 3.4.9). Weshall now prove that the Van Est Isomorphism is multiplicative if we endowthe spaces Ω∗(M,R)G and H∗c (G,R) with the ring structures provided bythe wedge product and cup product, respectively.We will prove this fact only in the case when G contains a cocompact lattice,as this allows us to give an easier argument.

Theorem 4.6.3. Let G be a connected Lie group and H a connected, compactsubgroup with G/H = M contractible. Suppose moreover that there exists acocompact lattice Γ < G. Let

V k : Ωk(M,R)→ C(Gk+1,R)

be the Van Est chain map defined in Remark 3.4.8 and inducing the isomor-phisms

V k : Ωk(M,R)G → Hkc (G,R).

Then the diagram

Ωk(M,R)G ⊗ Ωh(M,R)G Hkc (G,R)⊗Hh

c (G,R)

Ωk+h(M,R) Hk+hc (G,R)

∧

V k⊗V h

`

V p+q

commutes for each k, h ∈ N.

101

Proof. Let us denote by ZΩm(Γ\M,R) the set of closedm-forms on Γ\M andlet i : Γ → G be the inclusion map. Then, the following diagram commutes:

Ωm(M,R)Gp //

Vm

ZΩm(Γ\M,R)Im // Cm∞(Γ\M,R) = Cm∞(M,R)Γ

φ∗

C(Gm+1,R)G

i∗ // C(Γk+1,R)Γ ,

where the map φ∗ was defined in Theorem 4.2.9 upon choice of x ∈ M(which has to be the same point used to define V m) and the map Im is asin Theorem 4.3.7; the map p is simply

Ωm(M,R)G → Ωm(M,R)Γ ' Ωm(Γ\M,R)

and it takes values in ZΩm(Γ\M,R) since all G-invariant forms on M areclosed (see Theorem 3.4.9).Passing to the cohomology spaces, we get the commutative diagram:

Ωm(M,R)Gp //

Vm'

HmDR(Γ\M,R)

Im

'// Hm∞(Γ\M,R)

φ∗'

Hm(G,R) i∗ // Hm(Γ,R)

where we have used Theorems 3.4.9, 4.2.9, and 4.5.3 and Proposition 4.3.5.Now, let ω ∈ Ωk(M,R)G and η ∈ Ωh(M,R)G. To prove that(

V k(ω) ` V h(η))

= V k+h(ω ∧ η)

it is sufficient to show that

i∗(V k(ω) ` V h(η)

)= i∗

(V k+h(ω ∧ η)

).

Buti∗(V k(ω) ` V h(η)

)= i∗

(V k(ω)

)` i∗

(V h(η)

)=

= φ∗Ikp(ω) ` φ∗Ihp(η)

and

i∗(V k+h(ω ∧ η)

)= φ∗Ik+hp(ω ∧ η) = φ∗Ik+h (p(ω) ∧ p(η)) =

= φ∗(Ikp(ω) ∧ Ihp(η)

),

where the last equality follows from Theorem 4.3.7.Hence, to conclude the proof, we just have to check that the map φ∗ is

102

multiplicative. But this is clear already at the level of cochains: if f ∈Ck∞(M,R) and g ∈ Ch∞(M,R) and γ0, ..., γk+h ∈ Γ, then

(φ∗f ` φ∗g) (γ0, ..., γk+h) = (φ∗f)(γ0, ..., γk) · (φ∗g)(γk, ..., γk+h) =

= f (∆(γ0 · x, ..., γk · x)) · g (∆(γk · x, ..., γk+h · x)) ,

φ∗(f ` g)(γ0, ..., γk+h) = (f ` g) (∆(γ0 · x, ..., γk+h · x)) =

= f (∆(γ0 · x, ..., γk · x)) · g (∆(γk · x, ..., γk+h · x)) ,

where the last equality follows from the fact that, for straight simplices,

∆(γ0 · x, ..., γk+h · x)|[e0,...,ek] = ∆(γ0 · x, ..., γk · x) ,

∆(γ0 · x, ..., γk+h · x)|[ek,...,ek+h] = ∆(γk · x, ..., γk+h · x) .

103

104

Chapter 5

Gromov’s ProportionalityPrinciple

In Section 4.2 we defined the simplicial volume of a closed manifold in purelytopological terms; on the other hand, the volume of Riemannian manifoldsis a strictly geometric feature. In Section 5.1 we will prove Gromov’s cele-brated Proportionality Principle, stating that simplicial volume and Rieman-nian volume are proportional for closed manifolds with isometric universalcoverings.We will present Thurston’s proof of this important result (see [Thu78]), in-volving the definition of measure homology and the use of his smearing tech-nique. Despite both these ideas dating back to the early 80s, for a fullyformal exposition we will need to appeal to more recent results of Hansenand Zastrow (1998) and Löh (2006) establishing that, for smooth manifolds,the theories of measure homology and singular homology yield isometric re-sults.In Section 5.2 we will then show that the simplicial volume of a closed locallysymmetric space is nonzero if and only if it is of noncompact type, by ap-pealing to a result of Lafont and Schmidt ([LS06]). For symmetric spaces ofnoncompact type, we will furthermore characterise the proportionality con-stant in terms of the seminorm on the continuous cohomology spaces of theirisometry group; this result will be essential for the computation of the pro-portionality constant of H2×H2 in Chapter 6 and is due to Bucher-Karlsson([BK08a]).We shall also provide a wide variety of other examples of nonvanishing pro-portionality constant.

5.1 A proof of Gromov’s Proportionality Principle

We will start by defining the measure homology spaces of a smooth mani-fold; this will grant us more flexibility in the construction of fundamental

105

cycles. Indeed, we will not be forced to consider finite sums of simplices anymore (as in singular homology), allowing us to consider uncountable linearcombinations of simplices with infinitesimal coefficients, namely measures onthe space of (smooth) singular simplices.First, we will need the concept of signed measure.

Definition 5.1.1. A signed Borel measure on a topological space X is afunction on Borel sets µ : B(X,R)→ R such that

µ(E) =∑n≥0

µ(En)

for each Borel set E ⊆ X and each countable partition into Borel setsEnn∈N. In particular, we must have∑

n≥0

|µ(En)| <∞

for each Borel partition.

It is clear that the space of signed Borel measures on X is a vector space.The following classical result can be found, for example, in [Yos80].

Theorem 5.1.2 (Jordan decomposition/Hahn decomposition). A signedmeasure µ on X can be uniquely decomposed as µ = µ+ − µ− for finite(positive) measures µ+, µ− satisfying the additional requirement that thereexist disjoint Borel sets A,B with A tB = X and µ+(B) = 0, µ−(A) = 0.

We will call µ+ and µ− the positive and negative parts of µ. Moreover,we can define the finite (positive) measure |µ| := µ+ + µ−.

Definition 5.1.3. Let µ be a signed Borel measure on a topological spaceX. The support of µ is the closed subset of X:

Supp µ := x ∈ X | |µ|(U) > 0, ∀ open neigbourhood U of x .

Recall that in Section 4.3 we have defined the spaces C∞k (M,R) of smoothsingular chains. We can endow these spaces with the structure of Fréchetspace given by the C1-topology: simply proceed as in Example 1.5.4, buttake into account only the derivatives of orders 0 and 1.

Definition 5.1.4. The space of smooth measure chains on M is the setCk(M) of compactly supported signed Borel measures on C∞k (M,R) endowedwith the C1-topology.

Let ∂i : C∞k (M,R) → C∞k−1(M,R) be the map which associates to eachsimplex its i-th face. It is continuous with respect to the C1-topologies andso it defines a map:

(∂i)∗ : Ck(M)→ Ck−1(M)

106

where (∂i)∗µ(B) = µ(∂−1i (B)) for each Borel set B ⊆ C∞k−1(M,R).

We can now define a boundary map:

∂ : Ck(M)→ Ck−1(M)

µ 7→k∑i=0

(−1)i(∂i)∗µ

which clearly satisfies ∂2 = 0.

Definition 5.1.5. The measure homology spaces H∗(M) are defined as thehomology groups of the complex:

...∂ // Ck+1(M)

∂ // Ck(M)∂ // Ck−1(M)

∂ // ... .

We can endow the vector spaces Ck(M) with the norm

‖µ‖ := |µ|(C∞k (M,R)) ;

we then obtain a seminorm on Hk(M) by taking the infimum of ‖ · ‖ over allrepresentatives of a given classSmooth singular chains can be seen as elements of C∗(M) via the inclusion:

C∞k (M,R) → Ck(M)∑i

λi · σi 7→∑i

λi · δσi

which induces a natural map ι : H∞∗ (M,R)→ H∗(M).Moreover, we can consider the map

I : Ck(M)→(

Ωk(M,R))∗

given by

I(ν)(ω) :=

∫Ck(M,R)

(∫σω

)dν(σ) , ∀ν ∈ Ck(M), ∀ω ∈ Ωk(M,R) .

This value is well-defined since ν is compactly supported and the function

σ 7→∫σω

is continuous on Ck(M,R) endowed with the C1-topology.

Lemma 5.1.6. The map I is a chain map between the complex (C∗(M), ∂∗)and the dual complex of the De Rham complex.

107

Proof. Let ν ∈ Ck(M) and ω ∈ Ωk−1(M,R). Then:

I(∂ν)(ω) =k∑j=0

(−1)j ·∫Ck−1(M,R)

(∫σω

)d ((∂j)∗ν(σ)) =

=k∑j=0

(−1)j ·∫Ck(M,R)

(∫∂jσ

ω

)dν(σ) =

∫Ck(M,R)

(∫∂σω

)dν(σ) =

=

∫Ck(M,R)

(∫σdk−1ω

)dν(σ) = I(ν)(dk−1ω) .

So we obtain maps in cohomology:

I : Hk(M)→ Hk ((Ω∗(M,R))∗) '(HkDR(M,R)

)∗.

Observe that the composition

I ι : H∞k (M,R)→(HkDR(M,R)

)∗is always an isomorphism, being the dual of the map

Ik : HkDR(M,R)→ Hk

∞(M,R)

defined in Section 4.3.The following result is due to A. Zastrow ([Zas98]), S. K. Hansen ([Han98])and C. Löh ([L06]).

Theorem 5.1.7. For all smooth manifolds, the maps

ι : H∞k (M,R)→ Hk(M)

are isomorphisms for each k ≥ 0. Moreover, they are isometries if we con-sider the Gromov norm on H∞k (M,R) and the seminorm defined earlier onHk(M).

As a consequence, also the maps I : Hk(M) →(HkDR(M,R)

)∗ are iso-morphisms.We can now turn to the definition of Thurston’s smearing procedure.Let M be a connected, closed, oriented Riemannian manifold with univer-sal covering M ; let Γ < I+(M) be the group of covering automorphisms ofM →M . The group I+(M) is unimodular by Corollary 4.5.4 and we choosea Haar measure µ on I+(M), normalised so that µ

(Γ\I+(M)

)= 1.

108

If σ is a singular k-simplex in M , the action of I+(M) on Ck(M,R) definesa map

Φσ : I+(M)→ Ck(M,R)

f 7→ f σ .

Since we can canonically identify Ck(M,R) ' Γ\Ck(M,R), we obtain a map:

Φσ : Γ\I+(M)→ Ck(M,R) ,

which is continuous if we keep considering the C1-topology on Ck(M,R).The measure µ, induced by µ on Γ\G, can be pushed forward to a measure onCk(M,R) and, since Γ\I+(M) is compact, we obtain a compactly supported,finite, positive Borel measure.

Definition 5.1.8. The measure chain

smr(σ) := (Φσ)∗(µ) ∈ Ck(M)

will be called the smearing of the simplex σ.We can also define a smearing map

smr : Ck(M,R)→ Ck(M,R)

by extending linearly the definition of smr already given on singular sim-plices.

Lemma 5.1.9. The smearing map is a chain map. Moreover, given anyf ∈ I+(M), we have smr(f · σ) = smr(σ) for all singular simplices σ.

Proof. First, if g ∈ I+(M), denote by Rg the diffeomorphism of Γ\I+(M)

induced by the right translation Rg : I+(M)→ I+(M).It is clear that Φg·σ = Φσ Rg, so

smr(g · σ) = (Φg·σ)∗ µ = (Φσ)∗(Rg)∗µ = (Φσ)∗µ = smr(σ) ,

since µ is a right Haar measure on I+(M).To check that smr is a chain map, let σ ∈ Ck(M,R):

smr(∂σ) = smr

k∑j=0

(−1)j · ∂jσ

=

k∑j=0

(−1)j · smr(∂jσ) =

=k∑j=0

(−1)j · (Φ∂iσ)∗ µ =

k∑j=0

(−1)j · (∂i Φσ)∗ µ =

=

k∑j=0

(−1)j · (∂i)∗

(Φσ)∗µ = ∂ ((Φσ)∗µ) = ∂(smr(σ)) .

109

In particular, whenever N is another Riemannian manifold covered byM , we can consider the map

smrN,M : Ck(N,R)→ Ck(M) ,

which is also a chain map and hence induces a map

smrN,M : Hk(N,R)→ Hk(M) .

Assume now that N is compact and oriented as well (and that the orienta-tions induced on M by M and N coincide). Then N admits a fundamentalclass [N ] ∈ Hn(N,R) and we can consider the class smr([N ]) ∈ Hn(M).

Lemma 5.1.10. The class smr([N ]) represents vol(N)vol(M) · [M ] in Hn(M). Fur-

thermore, ‖smr([N ])‖ ≤ ‖N‖.

Proof. Due to the discussion preceding Theorem 5.1.7, to prove the firstassertion we just have to show that:

I(smr([N ]))(volM ) =vol(N)

vol(M)· (In)∗([M ])(volM ) ,

where volM is the volume form on M . If∑

i λi · σi is a smooth fundamentalclass for M , the right-hand side equals precisely:

vol(N)

vol(M)

∑i

λi ·∫σi

volM = vol(N)

by Corollary 4.3.6. If∑

i µi ·τi is a smooth fundamental class for N , we have

I(smr([N ]))(volM ) =∑i

µi · I(smr(τi))(volM ) =

=∑i

µi ·∫Cn(M,R)

(∫σvolM

)d(smr(τi)(σ)) =

=∑i

µi ·∫

Γ\I(M)volN (τi) dµ =

∑i

µi · volN (τi) = vol(N) .

Finally:smr([N ]) =

∑i

µi · smr(τi)

and, since ‖smr(τi)‖ = 1 for each i, we clearly have ‖smr([N ])‖ ≤∑

i |µi|and also ‖smr([N ])‖ ≤ ‖N‖, since the choice of the fundamental cycle on Nwas arbitrary.

We have finally obtained Gromov’s Proportionality Principle.

110

Corollary 5.1.11. Let M,N be connected, closed, oriented Riemannianmanifolds with isometric universal covering spaces. Then:

‖M‖vol(M)

=‖N‖vol(N)

.

Proof. By the previous lemma and Theorem 5.1.7:

‖M‖vol(M)

=

∥∥∥∥smr([N ])

vol(N)

∥∥∥∥ ≤ ‖N‖vol(N)

;

the opposite inequality is obtained by exchanging M and N .

Definition 5.1.12. For a connected Riemannian manifold M admitting alocally isometric covering M → M , with M a closed, orientable manifold,we say that the value

‖M‖vol(M)

∈ [0,+∞)

is the proportionality constant associated to M .

The machinery that we have just developed allows us to easily computethe proportionality constant for Hn, as we shall now show.We recall that the value

vn := supvol(σ) | σ is a geodesic n-simplex in Hn

is finite; see for example Lemma C.2.3 in [BP92]. We still need one lastingredient, i.e. a straightening procedure, which we shall now describe.If M is a nonpositively curved manifold and a singular simplex σ : ∆k →Mis given, we can consider a lift σ of σ to the universal covering M of M anddefine str(σ) as the straight simplex with the same vertices as σ (ordered inthe same way); the projection of str(σ) to a simplex in M does not dependon the chosen lift σ and hence it is appropriate to denote it by str(σ). Wewill say that str(σ) is the straightening of σ.Extending linearly str to C∗(M,R), we obtain a chain map which is obviouslynorm-non-increasing. Moreover, the map

str : H∗(M,R)→ H∗(M,R)

is the identity. Indeed, it is easy to prove that, at the level of chains, stris homotopic to the identity: one can construct a homotopy from each sim-plex in M to its straightening by following the geodesics, these homotopiesdescend to M and yield an algebraic homotopy thanks to the usual prismoperator.

Corollary 5.1.13. If Mn is closed, oriented and hyperbolic, then:

‖M‖vol(M)

=1

vn.

111

Proof. Let c be a fundamental cycle for M and set

c′ := str(c) =∑α

λα · σα .

We have vol(M) = volM (c) since, up to coboundaries (on which volM van-ishes), we can assume that the simplices in c are part of a triangulation ofM ; then:

vol(M) = volM (c) = volM (c′) =∑α

λα · vol(σα) ≤

≤ vn ·∑α

|λα| = vn · ‖c′‖1 ≤ vn · ‖c‖1

and, by the arbitrarity of c, we obtain:

‖M‖ ≥ vol(M)

vn.

Conversely, given ε > 0, choose a regular geodesic simplex σ in Hn so thatvol(σ) ≥ vn − ε; let moreover ρ be a reflection with respect to a hyperplaneof Hn. Consider the signed measure

µ := smr(σ)− smr(ρ σ) ∈ Cn(M)

and observe that, since for each i = 0, ..., n we can find an orientation-preserving isometry fi of Hn taking the vertices of ∂iσ to the vertices of∂i(ρ σ), we have fi ∂iσ = ∂i(ρ σ) and

smr(∂iσ)

= smr(∂i(ρ σ)

).

In particular, (∂i)∗µ = 0 for each i, so that µ is a cycle in Cn(M) and definesa class [µ] ∈ Hn(M). Now, proceeding just like in Lemma 5.1.10, we observethat:

[µ] =2vol(σ)

vol(M)· [M ] .

Since smr(σ) and smr(ρ σ) are measures with disjoint supports, it is clearthat ‖µ‖ = 2; this implies that

‖M‖ =

∥∥∥∥vol(M)

2vol(σ)· [µ]

∥∥∥∥ ≤ vol(M)

2vol(σ)· ‖µ‖ =

vol(M)

vol(σ)≤ vol(M)

vn − ε

and our assertion follows by the arbitrarity of ε.

Remark 5.1.14. Let M be a connected, nonpositively curved, closed, ori-ented Riemannian manifold with universal covering M . The first part of theproof of Lemma 5.1.13 actually shows us that, even in the present context:

‖M‖ ≥ vol(M)

C,

112

where

C := supvol(σ) | σ is a straight n-simplex in M ∈ (0,+∞] .

In particular, if C < ∞, then ‖M‖ > 0. This happens for all closed, neg-atively curved manifolds, as shown by Inoue and Yano in [IY81]. However,the same result does not hold for generic closed manifolds of nonpositivecurvature.

5.2 The proportionality constant for locally sym-metric spaces

As anticipated, we can show that the simplicial volume of locally symmetricspaces vanishes in several cases.

Proposition 5.2.1. Let M be a compact locally symmetric space. Then‖M‖ = 0 unless M is of noncompact type.

Proof. Let M be the universal covering space ofM and let Γ < I(M) be thegroup of automorphisms of the associated covering map. Since the simplicialvolume is well-behaved with respect to finite covering spaces (Lemma 4.2.6),we can assume that Γ < I0(M) =: G, also thanks to Lemma 1.4.5. Let n bethe dimension of M .The proof of Corollary 2.2.4 shows that we can decompose the universalcovering G of G as

G ' G0 ×G+ ×G− .Moreover, if Gp is the stabiliser of a point p ∈ M and Gp is the identitycomponent of its preimage in G, we saw that G/Gp ' G/Gp; in particular,Gp must contain the centre of G, since it coincides with the preimage of Gpin G.Furthermore, we have compact subgroups H0, H+, H− with Gp = H0×H+×H− and

M ' G0/H0 ×G+/H+ ×G−/H− .If M has a nontrivial component of compact type, the group G+ will benontrivial and compact. Hence, the compact subgroup

H := H0 ×G+ ×H− < G

satisfies dim G − dim H < dim M . Let H be the projection of H to G viathe covering map G→ G; it is a compact subgroup satisfying:

Ωn(G/H,R) = 0 .

Since H contains Gp and hence the centre of G, we have

G/H ' G/H ' G0/H0 ×G−/H−

113

and this space is contractible. Then, Theorem 3.4.7 shows that Hn(Γ) = 0and Remark 4.2.11 implies that ‖M‖ = 0.Hence, we are left with the case in which M has nonpositive curvature andM has an euclidean factor. It follows from the work of Eberlein (Corollary 2in [Ebe83]) that M is finitely covered by a manifold isometric to a productTk × N . Since tori have maps with degree bigger than 1, we deduce byLemma 4.2.3 that ‖Tk ×N‖ = 0 and ‖M‖ = 0.

On the other hand, it was shown by Lafont and Schmidt in [LS06] thatthe simplicial volume of a closed locally symmetric space of noncompacttype is always positive. The outline of the proof is similar to Remark 5.1.14,but they had to define a more complicated straightening procedure for thesespaces, such that the possible volumes of straight simplices are bounded fromabove even when there are planes with zero sectional curvature (with someexceptions).This fact, Remark 5.1.14 and Corollary 4.2.8 provide a good number ofmanifolds with nonvanishing simplicial volume. An even wider variety ofexamples can be obtained thanks to the following result due to Gromov (see§0.2 and §3.5 in [Gro82]).

Theorem 5.2.2. Let M,N be n-dimensional, connected, closed, orientablemanifolds. If n ≥ 3 we have:

‖M#N‖ = ‖M‖+ ‖N‖ .

Finally, we can give the following characterisation of the proportionalityconstant for symmetric spaces of noncompact type, due to Bucher-Karlsson([BK08a]). It also provides an alternative proof of the Proportionality Prin-ciple for symmetric spaces.

Proposition 5.2.3. Let Mn be a symmetric space of noncompact type andlet G be the identity component of its isometry group. Let v be the continu-ous n-cocycle constructed in Remark 3.4.12. Then, for each compact locallysymmetric space M covered by M , we have

vol(M)

‖M‖= ‖[v]‖ ,

where we are considering the seminorm on Hnc (G,R) defined in Section 4.1.

Proof. The automorphism group Γ of the covering map M → M sits inI(M) as a cocompact lattice. Since, thanks to Lemma 1.4.5, I(M) hasfinitely many connected components, the group Γ′ = Γ ∩G has finite indexin Γ and so corresponds to a finite cover M ′ of M . Since both the volumeand the simplicial volume are well-behaved with respect to finite coverings,

114

it is sufficient to prove the proposition for M ′.In this case, just like in Corollary 4.5.6, we have an isometric embedding

i : Hnc (G,R) → Hn(Γ′,R) ' Hn(M ′,R)

and, thanks to Lemma 4.2.7, we simply need to prove that

i([v]) = vol(M ′) · ωM ′ ,

with the notation of Definition 4.2.2.Consider the sequence of maps

C(Gn+1,R)G → C(Gn+1,R)Γ → C(Γ′n+1,R)Γ,

where the second arrow is the map induced by the inclusion Γ′n+1 → Gn+1.Their composition induces the morphism Hn

c (G,R) → Hn(Γ′,R) in coho-mology, as proven in Section 3.2. The class [v] is sent to the class [f ] ∈Hn(Γ′,R), where, given a base-point x0 ∈ M :

f : Γ′n+1 → R(γ0, ..., γn) 7→ vol

M(∆(γ0 · x0, ..., γn · x0))

and ∆(γ0 · x0, ..., γn · x0) denotes a straight simplex in M .Recalling Remark 4.3.4 and the proof of Theorem 4.2.9, where we definedthe map

(φn)∗ : Cn∞(M,R)→ Cn(Γ,R) ,

we see that [f ] = (φn)∗(c), where c is the cochain that associates to eachsmooth simplex its volume. The conclusion follows from Corollary 4.3.6.

Exploiting the previous proposition, we can give one more proof of Corol-lary 5.1.13, adopting a dual point of view. We remark this especially becausethe computation of the proportionality constant for H2 × H2 which we willcarry out in Chapter 6 is in the same spirit.

Alternative proof of Corollary 5.1.13. Let G be the connected group I+(Hn)and let H be the stabiliser of a point in Hn. We will see in Chapter 6 thatHn is a symmetric space of noncompact type (Remark 6.1.6); then, thanksto Proposition 5.2.3, it suffices to prove that ‖[v]‖ = vn.The results of Section 4.4 imply that the maps H∗cb(G,R) → H∗c (G,R) areinduced by the inclusion:

Cb,alt((Hn)∗+1,R)G → Calt((Hn)∗+1,R)G ;

in addition, the complex Cb,alt((Hn)∗+1,R)G induces the canonical seminormon H∗cb(G,R).The class [v] ∈ Hn

c (G,R) is represented by the alternating function:

ν : (Hn)n+1 → R(x0, ..., xn) 7→ volHn

(∆(x0, ..., xn)

)115

and, since ‖ν‖∞ = vn, it is clear that ‖[v]‖ ≤ vn. To show that this is anequality we need to prove that, for each alternating, G-invariant, continuousmap f : (Hn)n → R, we have ‖ν + δf‖∞ ≥ vn.Observe that, if (y0, ..., yn−1) are the vertices of a geodesic, regular (n− 1)-simplex in Hn, then f(y0, ..., yn−1) = 0. Indeed, there exists g ∈ G swappingy0 and y1 and fixing all the other points; g can be constructed by composing asuitable reflection within the hyperplane containing the yi with the reflectionwith respect to the same hyperplane. Then, since f is alternating and G-invariant, it must satisfy:

f(y0, ..., yn−1) = f(g · y0, ..., g · yn−1) = −f(y0, ..., yn−1) .

Let now R > 0 and let σR be a geodesic, regular simplex in Hn with edgesof length R; call yR0 , ..., yRn its vertices. The previous remark shows thatδf(yR0 , ..., y

Rn ) = f

(∂(yR0 , ..., y

Rn ))

= 0. So:

(ν + δf)(yR0 , ..., yRn ) = ν(yR0 , ..., y

Rn ) = vol(σR) .

We can choose the σR so that their vertices converge to hose of a regular idealsimplex, for R→ +∞. By a well-known result of Haagerup and Munkholm([HM81]) the volume of all such simplices is vn. In particular, vol(σR)→ vnand we obtain ‖ν + δf‖∞ ≥ vn.

116

Chapter 6

Manifolds Covered By H2 ×H2

In this chapter we will compute the proportionality constant for H2×H2, fol-lowing Bucher-Karlsson ([BK08b]). As we shall see, H2×H2 is a symmetricspace of noncompact type, so Proposition 5.2 in the previous chapter ex-presses this value in terms of the seminorm of a class in H4

c (I0(H2×H2),R).We will calculate this seminorm by relating it to the seminorm of a class inthe continuous bounded cohomology of the full isometry group of H2 × H2

and finding a suitable representative.Then, in Section 6.5, we will show that the proportionality constant forH2×H2 is not the supremum of values of straight simplices. This is a notabledifference from the only other case in which the proportionality constant isknown and nonzero, i.e. hyperbolic n-space.

6.1 Sectional curvature of H2 ×H2

Let M,N be Riemannian manifolds and consider their Riemannian productM ×N , with projections π1 : M ×N →M and π2 : M ×N → N .If X is a vector field on M , we can define the vector field π∗1X on M ×N asthe only vector field that is tangent to all the leaves M × ∗ and is takento X by dπ1. Similarly, we can define π∗2Y for each vector field Y on N .We recall the following easy results.

Lemma 6.1.1. Let X1, ..., Xm be an orthonormal frame on some open setU ⊆ M and Y1, ..., Yn an orthonormal frame on V ⊆ N ; define the vec-tor fields Xi := π∗1Xi and Yi := π∗2Yi. Let ∇M ,∇N ,∇ be the Levi-Civitaconnections on M, N, M ×N , respectively. Then:

• [Xi, Yj ] = 0, [Xi, Xj ] = π∗1[Xi, Xj ], [Yi, Yj ] = π∗2[Yi, Yj ];

• ∇XiXj = π∗1

(∇MXiXj

), ∇

YiYj = π∗2

(∇NYiYj

);

• ∇XiYj = 0, ∇

YiXj = 0.

117

Corollary 6.1.2. In the setting of the previous lemma, let Z1, ..., Z4 be vectorfields on U × V , chosen among X1, ..., Xm, Y1, ..., Yn and let R be the cur-vature (0, 4)-tensor on M × N . Then R(Z1, Z2)Z3 = 0, unless Z1, Z2, Z3

were all chosen among the Xi or all among the Yi. As a consequence,R(Z1, Z2, Z3, Z4) = 0, unless Z1, Z2, Z3, Z4 were all chosen among the Xi

or all among the Yi.

Corollary 6.1.3. The sectional curvature of each tangent plane to H2×H2

lies in the interval [−1, 0]. The only planes with sectional curvature −1 arethe tangent planes of the leaves H2 × ∗ and ∗ ×H2.

Proof. Since H2 × H2 is homogeneous, we can restrict our attention to theplanes contained in T(p,p)(H2 × H2), for some p ∈ H2. Let u, v be an or-thonormal basis of TpH2 ⊕ 0 ⊆ T(p,p)(H2 × H2) and w, z an orthonormalbasis of 0 ⊕ TpH2.Let a, b be an orthonormal basis of some other plane in T(p,p)(H2 ×H2); wecan write

a = α1u+ α2v + α3w + α4z,

b = β1u+ β2v + β3w + β4z,

for some orthonormal vectors α, β ∈ R4. Due to Corollary 6.1.2, it is easyto check that

R(a, b, a, b) = (α1β2 − α2β1)2R(u, v, u, v) + (α3β4 − α4β3)2R(w, z, w, z) =

= −(α1β2 − α2β1)2 − (α3β4 − α4β3)2.

Defining α′ = (α1, α2, 0) ∈ R3, α′′ = (α3, α4, 0) ∈ R3 and similarly β′, β′′, wehave

0 ≤ −R(a, b, a, b) = |α′ ∧ β′|2 + |α′′ ∧ β′′|2 ≤ |α′|2 · |β′|2 + |α′′|2 · |β′′|2 =

= |α′|2 · |β′|2 +(1− |α′|2

)·(1− |β′|2

)≤ 1,

and R(a, b, a, b) = −1 is possible only if α′ = β′ = 0 or α′′ = β′′ = 0.

Observe that, given f, g ∈ I(H2), we can define an isometry of H2 ×H2:

(f, g) : H2 ×H2 → H2 ×H2

(x, y) 7→ (f(x), g(y)) .

In particular, we obtain a subgroup I(H2)× I(H2) < I(H2 ×H2).

Corollary 6.1.4. The subgroup I(H2)× I(H2) < I(H2 ×H2) has index 2.

118

Proof. Define the isometry

τ : H2 ×H2 → H2 ×H2

(x, y) 7→ (y, x) .

Let f be an isometry of H2 ×H2 that does not lie in I(H2)× I(H2). SinceI(H2)×I(H2) acts transitively on H2 ×H2, we can find g ∈ I(H2)×I(H2)such that g f fixes a point (p, p) ∈ H2 × H2. By the previous corollary,d(p,p)(g f) can either preserve both the planes TpH2 ⊕ 0, 0 ⊕ TpH2 orexchange them. Observe that we must be in the latter case, otherwise g fwould coincide with some element of I(H2)×I(H2) at the first order in (p, p)and we would therefore have g f ∈ I(H2)× I(H2) (see Lemma 1.4.1).The same argument shows nonetheless that τ gf must lie in I(H2)×I(H2).So, for some g′, g′′ ∈ I(H2)× I(H2) we have

f = g−1τg′ = τg′′g′,

since τ clearly lies in the normaliser of I(H2)× I(H2).We conclude that:

I(H2 ×H2) =(I(H2)× I(H2)

)∪ τ

(I(H2)× I(H2)

).

Finally, we recall the following easy consequence of Lemma 6.1.1.

Lemma 6.1.5. Let M,N be Riemannian manifolds and γM : [0, 1] → Mand γN : [0, 1]→ N two geodesic segments. Then

(γM , γN ) : [0, 1]→M ×Nt 7→ (γM (t), γN (t))

is a geodesic segment in M ×N .

Remark 6.1.6. Let M be a symmetric space with strictly negative curva-ture. In the decomposition yielded by Corollary 2.2.4 two out of the threefactors must be trivial. Indeed, if it wereM = M1×M2 with bothM1,M2 ofdimension at least 1, each product Σ of a geodesic arc in M1 and a geodesicarc in M2 would be a totally geodesic flat surface, as is easy to check usingLemma 6.1.5; the tangent spaces of Σ would then be planes with curvaturezero in M , due to Proposition 1.4.10, and this is absurd.Hence Theorem 2.3.2 shows that a symmetric space with stricly negative cur-vature is always of noncompact type; similarly, positively curved symmetricspaces are of compact type.As a consequence, Hn is a symmetric space of noncompact type for eachn ≥ 2.

119

Remark 6.1.7. A product of symmetric spaces of noncompact type is againa symmetric space of noncompact type. This follows from the fact that, ifg = h ⊕ p is a Cartan decomposition for a semisimple, real Lie algebra g,then g⊕ g = (h⊕ h)⊕ (p⊕ p) is a Cartan decomposition for g⊕ g.As a consequence, H2 ×H2 is a symmetric space of noncompact type.

6.2 The isometry group of the hyperbolic plane

Throughout this section, let G := I+(H2).Recall that H2

c (G,R) ' R thanks to Corollary 3.4.11. Given x ∈ H2, Re-mark 3.4.12 provides a cocycle vx ∈ C(G3,R)G representing a non-zero classH2c (G,R); call ωH2 this class, which is clearly independent of the choice of

the point x. We recall that vx is given by

vx(g0, g1, g2) = volH2

(∆(g0 · x, g1 · x, g2 · x)

),

where ∆(g0 ·x, g1 ·x, g2 ·x) is simply the geodesic triangle in H2 with verticesg0 · x, g1 · x, g2 · x; we remark that, according to our conventions, volH2 isnegative on negatively oriented simplices.We can see that vx ∈ Cb(G3,R)G and ‖vx‖∞ = π, so the class ωH2 must liein the image of the comparison morphism ι2; let ωxb,H2 ∈ H2

cb(G,R) be thecontinuous bounded cohomology class defined by vx.Choose a point ξ ∈ ∂H2 and consider the function

Orξ : G3 → R(g0, g1, g2) 7→ Or(g0 · ξ, g1 · ξ, g2 · ξ) ,

where the function Or : (∂H2)3 → R is defined as

Or(ξ0, ξ1, ξ2) =

+1, if ξ0, ξ1, ξ2 are positively oriented;−1, if ξ0, ξ1, ξ2 are negatively oriented;0, if ξ0, ξ1, ξ2 are not mutually distinct.

The function Orξ is Borel measurable and lies in the space L∞alt(G3,R), con-

sidering the measure class of the counting measure on G3.

Lemma 6.2.1. 1. The class ωxb,H2 ∈ H2cb(G,R) does not depend on the

choice of x; we will denote it simply by ωb,H2.

2. For each ξ ∈ ∂H2, the function π · Orξ ∈ L∞(G3,R)G is a cocyclerepresenting the class ωb,H2 .

Proof. Given points p, q ∈ H2, we define the function bp,q : G3 → R as follows:

bp,q(g0, g1, g2) := vol(∆(g1 · p, g2 · p, g2 · q)

)− vol

(∆(g1 · p, g1 · q, g2 · q)

)+

− vol(∆(g0 · p, g2 · p, g2 · q)

)+ vol

(∆(g0 · p, g0 · q, g2 · q)

)+

+ vol(∆(g0 · p, g1 · p, g1 · q)

)− vol

(∆(g0 · p, g0 · q, g1 · q)

).

120

It is easy to check that bp,q is Borel measurable and ‖bp,q‖∞ ≤ 6π; further-more, bp,q ∈ L∞(G3,R)G.To prove the first part of the lemma, let x and y be points in H2; it suffices tonotice that vx− vy = dbx,y. This follows immediately from Stokes’ Theoremand the fact that the singular 2-chain

∆(g0 · x, g1 · x, g2 · x)−∆(g0 · y, g1 · y, g2 · y)+

−∆(g1 · x, g2 · x, g2 · y) + ∆(g1 · x, g1 · y, g2 · y)+

+ ∆(g0 · x, g2 · x, g2 · y)−∆(g0 · x, g0 · y, g2 · y)+

−∆(g0 · x, g1 · x, g1 · y) + ∆(g0 · x, g0 · y, g1 · y)

is the boundary of the following 3-chain:

−∆(g0·x, g0·y, g1·y, g2·y)+∆(g0·x, g1·x, g1·y, g2·y)−∆(g0·x, g1·x, g2·x, g2·y) .

To prove the second part of the lemma, one has to show that, if x ∈ H2 andξ ∈ ∂H2, then vx − π ·Orξ = dbx,ξ and appeal to Corollary 4.4.15. Observethat:

π ·Orξ(g0, g1, g2) = vol(∆(g0 · ξ, g1 · ξ, g2 · ξ)

).

We can then proceed essentially as in the proof of the first part of the lemma.The only point that needs some care is the use of Stokes’ Theorem: one needsto choose small horospheres around the involved ideal points and use themto carve out from the involved simplices a neighbourhood of all the verticesthat lie on ∂H2.

Lemma 6.2.2. The subgroup Gξ of G fixing ξ ∈ ∂H2 is amenable.

Proof. Since G acts transitively on ∂H2 we can assume that ξ = ∞ in thehalf-plane model. The linear fractional transformations fixing ∞ are pre-cisely those of the form

z 7→ λz + b, λ ∈ R+, b ∈ R.

Hence we can identify H with the group of orientation-preserving, affinetransformations of the real line; in particular H = T o D, where T is thesubgroup of translations z 7→ z+ b, b ∈ R and D is the subgroup of dilationsz 7→ λz, λ ∈ R+. So H is solvable because its commutator subgroup isH ′ = [H,H] = T and H(2) = [T, T ] = e. Solvable groups are amenable;see Corollary 13.5 in [Pie84].

Since G acts transitively on ∂H2, we can identify G/Gξ ' ∂H2 ' S1. SoCorollary 4.4.23 yields the following result.

Corollary 6.2.3. The complex

0 L∞alt(S1,R)G L∞alt(S

1 × S1,R)G ...d0 d1

121

computes the bounded cohomology spaces Hkcb(G,R) and induces the canonical

seminorm on them.

Hence the function π·Or : (S1)3 → R, being an element of L∞alt((S1)3,R)G,

induces a cohomology class in H2cb(G,R) and it is clear that this class must

be precisely ωb,H2 (see Propositions 4.4.13 and 4.4.22).

6.3 The isometry group of H2 ×H2

Throughout this section, define G := I0(H2 ×H2) = I+(H2)× I+(H2) andH := I(H2 ×H2). We will denote by π1, π2 the continuous homomorphismsprojecting G onto the two factors I+(H2). Due to Corollary 6.1.4, the groupI(H2)× I(H2) has index 2 in H, so G has index 8 in H.The class ωH2 ∈ H2

c (I+(H2),R) defines classes π∗1ωH2 , π∗2ωH2 in H2c (G,R)

and we can consider their cup product

ωH2×H2 := π∗1ωH2 ` π∗2ωH2 ∈ H4c (G,R).

Observe that G has compact subgroups of the form K ' S1 × S1, suchthat G/K ' H2 × H2; hence we can exploit Van Est’s Theorem and Re-mark 3.4.12. In the following discussion, we will use the notation π1, π2 alsofor the projections of H2 ×H2 onto its factors.

Lemma 6.3.1. If volH2 is the Riemannian volume form of H2 and V denotesthe Van Est Isomorphisms for G, we have

V 2(π∗i volH2) = π∗i ωH2 , for i = 1, 2.

Proof. Fix some p ∈ H2 and consider the cocycle vp ∈ C((I+(H2))3,R)representing ωH2 . Let g0, g1, g2 be elements of G and define the isometriesfj := π1(gj) ∈ I+(H2) for j = 0, 1, 2. We will the prove the lemma only fori = 1, the other case being identical.A cocyle representing π∗1ωH2 is given by:

(vp π1)(g0, g1, g2) =

∫∆(f0(p),f1(p),f2(p))

volH2 ,

whereas a cocyle representing V 2(π∗i volH2) is:

(g0, g1, g2) 7→∫

∆(g0·(p,p), g1·(p,p), g2·(p,p)

) π∗1volH2 =

=

∫π1(

∆(g0·(p,p), g1·(p,p), g2·(p,p)

)) volH2 = (vp π1)(g0, g1, g2).

122

Proposition 6.3.2. If volH2×H2 is the volume form of H2 ×H2, we have

V 4(volH2×H2) = ωH2×H2 .

Proof. SincevolH2×H2 = π∗1volH2 ∧ π∗2volH2 ,

we haveV 4(volH2×H2) = V 4 (π∗1volH2 ∧ π∗2volH2) =

= V 2(π∗1volH2) ` V 2(π∗2volH2) = π∗1ωH2 ` π∗2ωH2 = ωH2×H2 ,

where the second equality follows from Theorem 4.6.3 and the third fromthe previous lemma.

Since H4c (G,R) ' R by Corollary 3.4.11, we conclude that the class

ωH2×H2 generates H4c (G,R). Moreover, Proposition 6.3.2 shows that ωH2×H2

is precisely the class whose seminorm appears in the statement of Proposi-tion 5.2; our aim is now to compute ‖ωH2×H2‖.If we set ωb,H2×H2 := π∗1ωb,H2 ` π∗2ωb,H2 , we must have

ι4(ωb,H2×H2) = ωH2×H2 .

Lemma 6.2.2 and Remark 4.4.12 imply that, for ξ ∈ ∂H2, the subgroupGξ,ξ < G fixing (ξ, ξ) ∈ ∂H2 × ∂H2 is amenable. The quotient G/Gξ,ξcan be identified with ∂H2 × ∂H2 ' S1 × S1; hence, Corollary 4.4.15 andCorollary 4.4.23 have the following consequence:

Corollary 6.3.3. The complexes

0 L∞(S1 × S1,R)G L∞((S1 × S1)2,R)G ... ,

0 L∞alt(S1 × S1,R)G L∞alt((S

1 × S1)2,R)G ... ,

d0 d1

d0 d1

compute the bounded cohomology spaces Hkcb(G,R) and induce the canonical

seminorm on them.

Now we can find a suitable representative for the class ωb,H2×H2 . Denoteby π1 and π2 also the projections of S1 × S1 ' ∂H2 × ∂H2 onto its factors.

Lemma 6.3.4. The function π2 · (π∗1Or ` π∗2Or) ∈ L∞((S1 × S1)5,R)G

represents the class ωb,H2×H2 . The same holds for the function:

Θ := π2 · Alt(π∗1Or ` π∗2Or) ∈ L∞alt((S1 × S1)5,R)G.

123

Proof. We have seen in Section 6.2 that π · Or ∈ L∞((S1)3,R)I+(H2) rep-

resents the class ωb,H2 ∈ H2cb(I+(H2),R). Hence, for i = 1, 2, the functions

π · π∗iOr ∈ L∞((S1 × S1)3,R)G represent the classes π∗i ωb,H2 ∈ H2cb(G,R).

Indeed, it is easy to check that the following diagram commutes:

Cb((I+(H2)

)3,R)

Cb((I+(H2)× I+(H2)

)3,R)

L∞((I+(H2)/A

)3,R)

L∞((I+(H2)/A× I+(H2)/A

)3,R),

π∗i

τ2 τ2

π∗i

where A is the stabiliser of some ξ ∈ ∂H2 in I+(H2) and the maps τ weredefined in the proof of Proposition 4.4.13; here, given a left invariant meanon A, we are endowing A × A with the left invariant mean constructed inRemark 4.4.12.The fact that π2 · (π∗1Or ` π∗2Or) represents ωb,H2×H2 then follows fromRemark 4.6.2. The function Θ represents the same class because of Propo-sition 4.4.22.

Now let R be the H-module that is isomorphic to R as a vector spaceand on which each h ∈ H acts by the multiplication by ±1, depending onwhether or not h preserves the orientation of H2×H2. Clearly, if i : G → His the inclusion map, we have that i∗R is the trivial G-module R.We are about to show that ‖ωH2×H2‖ coincides with the seminorm of a classωHH2×H2 ∈ H4

c (H, R). This will prove useful since it will allow us to work withmaps on (S1 × S1)k which are H-invariant; this is a considerably strongerrequirement than simple G-invariance.Fix a reflection ρ for H2. A set of representatives for the cosets of G in H isgiven by

R = id, σ1, σ2, σ1σ2, τ, σ1τ, σ2τ, σ1σ2τ,

where

σ1 : H2 ×H2 → H2 ×H2 σ2 : H2 ×H2 → H2 ×H2

(x, y) 7→ (ρ(x), y) (x, y) 7→ (x, ρ(y)) .

and the map τ was defined in the proof of Corollary 6.1.4. Since σ1, σ2, τare involutions, observing that in addition σ1σ2 = σ2σ1 and τσ1 = σ2τ ,τσ2 = σ1τ , it is easy to check that R is a subgroup of H.

Proposition 6.3.5. The maps

i∗ : H∗cb(H, R)→ H∗cb(G,R), i∗ : H∗c (H, R)→ H∗c (G,R)

are isometric embeddings.

124

Proof. Thanks to Corollary 4.4.3 it suffices to construct left inverses (norm-non-increasing ones in the bounded case) for:

i∗ : Cb((H2 ×H2)∗+1, R)H → Cb((H2 ×H2)∗+1,R)G,

i∗ : C((H2 ×H2)∗+1, R)H → C((H2 ×H2)∗+1,R)G.

We will exhibit chain maps

trans∗b : Cb((H2 ×H2)∗+1,R)G → Cb((H2 ×H2)∗+1, R)H ,

trans∗ : C((H2 ×H2)∗+1,R)G → C((H2 ×H2)∗+1, R)H ,

such that trans∗b i∗ = id, trans∗ i∗ = id, thus proving that i∗ is injectivein cohomology. Moreover, the induced transfer morphisms will fit into thecommutative diagram

Hkc (H, R)

i∗ // Hkc (G,R)

trans // Hkc (H, R)

Hkcb(H, R)

i∗ //

ιk

OO

Hkcb(G,R)

ιk

OO

transb // Hkcb(H, R)

ιk

OO

and transb will be 1-Lipschitz; since i∗ is also 1-Lipschitz, we will concludethat it is isometric, both in cohomology and bounded cohomology.Define, for f ∈ C((H2 ×H2)k+1,R)G:

(transkf)(x0, ..., xk) :=1

8

∑r∈R

εr · f(r · x0, ..., r · xk),

where εr = 1 if r is orientation-preserving and εr = −1 otherwise. For h ∈ H,we can find r ∈ R such that h = gr−1 with g ∈ G; then:

(h · transkf)(x0, ..., xk) = εr · (transkf)(h−1 · x0, ..., h−1 · xk) =

=εr8

∑r∈R

εr ·f(rh−1 ·x0, ..., rh−1 ·xk) =

εr8

∑r∈R

εr ·f(rrg−1 ·x0, ..., rrg−1 ·xk) =

=ε2r8εr ·∑r∈R

εr · f(rg−1 · x0, ..., rg−1 · xk) =

1

8·∑r∈R

εr · f(r · x0, ..., r · xk) =

= (transkf)(x0, ..., xk),

where the second-last equality is due to the fact that, G being normal in H,for each r ∈ R there exists gr ∈ G with rg−1 = grr and so:

f(rg−1 · x0, ..., rg−1 · xk) = f(grr · x0, ..., grr · xk) = f(r · x0, ..., r · xk) ,

due to the G-invariance of f . Therefore, we have proven that transkf isH-invariant.

125

Furthermore, it is easy to see that trans is a chain map taking boundedmaps to bounded maps; this allows us to define transb and it is clear that‖transb‖ ≤ 1. Finally, if f is already H-invariant, we have transkf = f , thusshowing that trans∗ i∗ and trans∗b i∗ are the identity.

Now observe that H acts on ∂H2 × ∂H2 ' S1 × S1 and the functionΘ is also H-invariant. Since the stabiliser of (ξ, ξ) in H contains Gξ,ξ as afinite-index, normal subgroup, we conclude that it is amenable as well; seefor example Proposition 13.4 in [Pie84].Then, by Corollary 4.4.23, Θ ∈ L∞alt((S

1 × S1)5,R)H defines a continuousbounded cohomology class

ωHb,H2×H2 ∈ H4cb(H, R).

We also define ωHH2×H2 := ι4(ωHb,H2×H2) ∈ H4c (H, R). The relations

i∗(ωHb,H2×H2) = ωb,H2×H2 ,

i∗(ωHH2×H2) = ωH2×H2 ,

are then obvious since i∗ : H4cb(H, R)→ H4

cb(G,R) is induced by the inclusion

L∞alt((H2 ×H2)5, R)H → L∞alt((H2 ×H2)5,R)G

and the comparison maps commute with i∗.In Section 6.4 we will prove the following result, due to Bucher-Karlsson([BK08b]).

Theorem 6.3.6. ‖ωHb,H2×H2‖ = 2π2/3.

Moreover the following holds:

Theorem 6.3.7. The comparison map

ι4 : H4cb(H, R)→ H4

c (H, R)

is an isomorphism.

The proof of this fact is particularly cumbersome. It suffices to provethat ι4 is injective since H4

c (H, R) ' R by Corollary 3.4.11 and Proposi-tion 6.3.5. This is achieved by an inductive procedure that, given a Borelfunction f : (S1 × S1)5 → R with certain properties, constructs a bounded,alternating, H-invariant Borel function g : (S1×S1)4 → R with f = d3g. Fora detailed treatment we refer the reader to Bucher-Karlsson’s paper [BK08b].The next corollaries easily follow from the previous results.

Corollary 6.3.8. ‖ωH2×H2‖ = 2π2/3

126

Proof. Theorems 6.3.7 and 6.3.6 immediately give: ‖ωHH2×H2‖ = 2π2/3. Thenit suffices to apply Proposition 6.3.5.

Corollary 6.3.9. If M is a connected, closed, oriented manifold covered byH2 ×H2 we have

‖M‖vol(M)

=3

2π2.

Proof. Thanks to Proposition 5.2.3, we have

‖M‖vol(M)

=1

‖ωH2×H2‖.

Now it suffices to apply the previous corollary.

6.4 Proof of Theorem 6.3.6

In this section, we will compute the value that the seminorm induced by thecomplex

(L∞alt((S

1 × S1)∗+1, R)H , d∗)takes on the class

1

π2[Θ] = [Alt(π∗1Or ` π∗2Or)] ∈ H4

cb(H,R) .

This will consist of two steps, namely Lemma 6.4.1 and Proposition 6.4.2.Theorem 6.3.6 is an immediate consequence of these results.

Lemma 6.4.1. ‖Alt(π∗1Or ` π∗2Or)‖∞ = 2/3.

Proposition 6.4.2. For each f ∈ L∞alt((S1 × S1)4, R)H , we have

‖d3f + Alt(π∗1Or ` π∗2Or)‖∞ ≥2

3.

Throughout this section we will consider the measure class of the countingmeasure on S1×S1. In particular, the elements of L∞alt((S

1×S1)k+1, R) areBorel functions that are defined everywhere.

Proof of Lemma 6.4.1. Choose points zi = (xi, yi) ∈ S1 × S1 for i = 0, ..., 4.Then:

Alt(π∗1Or ` π∗2Or)(z0, ..., z5) =

=1

120

∑σ∈S5

εσ ·Or(xσ(0), xσ(1), xσ(2)) ·Or(yσ(2), yσ(3), yσ(4)).

We want to show that the module of this value is bounded from above by2/3 and that it is exactly 2/3 for appropriately chosen zi.Define τ = (01234) and

Ω := σ ∈ S5 | σ(2) = 0;

127

this is a set of cardinality 24 and each σ ∈ S5 can be written as σ = τk α,with k = 0, ..., 4 and α ∈ Ω, in a unique way. Observe that Ω can bepartitioned in 6 sets

Ωi,j := σ ∈ Ω | σ(0), σ(1) = i, j

for i, j = 1, ..., 4.Recalling that Or is alternating, we have

Alt(π∗1Or ` π∗2Or)(z0, ..., z5) =

=1

120

4∑k=0

∑α∈Ω

ετkα·Or(xτkα(0), xτkα(1), xτkα(2))·Or(yτkα(2), yτkα(3), yτkα(4)) =

=1

120

4∑k=0

∑i,j

∑α∈Ωi,j

εα·Or(xτkα(0), xτkα(1), xτkα(2))·Or(yτkα(2), yτkα(3), yτkα(4)),

and, observing that for all α ∈ Ωi.j the expression

εα ·Or(xτkα(0), xτkα(1), xτkα(2)) ·Or(yτkα(2), yτkα(3), yτkα(4))

takes the same value, we obtain

Alt(π∗1Or ` π∗2Or)(z0, ..., z5) =

=1

30

4∑k=0

[Or(xτk(0), xτk(1), xτk(2)) ·Or(yτk(0), yτk(3), yτk(4))+

−Or(xτk(0), xτk(1), xτk(3)) ·Or(yτk(0), yτk(2), yτk(4))+

+Or(xτk(0), xτk(1), xτk(4)) ·Or(yτk(0), yτk(2), yτk(3))+

+Or(xτk(0), xτk(2), xτk(3)) ·Or(yτk(0), yτk(1), yτk(4))+

−Or(xτk(0), xτk(2), xτk(4)) ·Or(yτk(0), yτk(1), yτk(3))+

+Or(xτk(0), xτk(3), xτk(4)) ·Or(yτk(0), yτk(1), yτk(2))].

We now divide the proof into two cases.

Case 1: the xi are all distinct.Since Alt(π∗1Or ` π∗2Or) is alternating, we can permute the zi so thatthe xi are cyclically ordered in S1, without changing the absolute valueof Alt(π∗1Or ` π∗2Or)(z0, ..., z5).Then,

Or(xτ(i), xτ(j), xτ(k)) = Or(xi, xj , xk), ∀i, j, k = 0, ..., 4

128

andAlt(π∗1Or ` π∗2Or)(z0, ..., z5) =

=1

30

4∑k=0

[Or(yτk(0), yτk(3), yτk(4))−Or(yτk(0), yτk(2), yτk(4))+

+Or(yτk(0), yτk(2), yτk(3)) +Or(yτk(0), yτk(1), yτk(4))+

−Or(yτk(0), yτk(1), yτk(3)) +Or(yτk(0), yτk(1), yτk(2))].

Now we substitute the cocyle relation d3Or = 0, i.e.

Or(yτk(0), yτk(1), yτk(2)) +Or(yτk(0), yτk(2), yτk(3))+

−Or(yτk(0), yτk(1), yτk(3)) = Or(yτk(1), yτk(2), yτk(3)).

We obtain:Alt(π∗1Or ` π∗2Or)(z0, ..., z5) =

=1

30

4∑k=0

[Or(yτk(0), yτk(3), yτk(4))−Or(yτk(0), yτk(2), yτk(4))+

+Or(yτk(1), yτk(2), yτk(3)) +Or(yτk(0), yτk(1), yτk(4))].

Since the summation involves only 20 terms and the absolute value ofeach of them is at most 1, we immediately get

|Alt(π∗1Or ` π∗2Or)(z0, ..., z5)| ≤ 20/30 = 2/3.

Moreover, it is easy to check that if the points wi ∈ S1 are positivelyoriented:

Alt(π∗1Or ` π∗2Or)(

(x0, w0), (x1, w2), (x2, w4), (x3, w1), (x4, w3))

= 2/3.

Case 2: : the xi are not all distinct.We can assume that also the yi are not all distinct, otherwise we couldproceed as in Case 1. Up to permuting the zi, we can assume that x0 =x1 and either y0 = y1, y0 = y2 or y2 = y3. We will study these subcasesseparately. Nevertheless, observe that, in all three subcases, at least 9of the 30 terms in the formula for Alt(π∗1Or ` π∗2Or)(z0, ..., z5) vanishsince they contain a factor Or(x0, x1, ·). Hence, we just have to findone additional vanishing summand to show

|Alt(π∗1Or ` π∗2Or)(z0, ..., z5)| ≤ 20/30 = 2/3.

Subcase 1: y0 = y1.In this case z0 = z1 and Alt(π∗1Or ` π∗2Or) must vanish on(z0, ..., z5) since it is alternating.

129

Subcase 2: y0 = y2.Consider the summand Or(x0, x3, x4) · Or(y0, y1, y2) = 0, whichappears for k = 0.

Subcase 3: y2 = y3.Consider the summand:

Or(xτ2(0), xτ2(2), xτ2(3)) ·Or(yτ2(0), yτ2(1), yτ2(4)) =

= Or(x2, x4, x0) ·Or(y2, y3, y1) = 0.

Before proving Proposition 6.4.2, we will need to establish an easy lemma.

Lemma 6.4.3. Let f ∈ L∞alt((S1×S1)4, R)H . Let x0, x1, x2, x3 and y0, y1, y2, y3

be points in S1. Suppose that the geodesics x0x2 and x1x3 in H2 share a com-mon point, while the geodesics y0y2 and y1y3 are disjoint.Then:

f(

(x0, y0), (x1, y1), (x2, y2), (x3, y3))

= 0.

Proof. Let p = x0x2 ∩ x1x3 and let F be the geodesic symmetry of H2 atp. Clearly, F exchanges x0 with x2 and x1 with x3.Let γ be the unique geodesic in H2 which is orthogonal to both y0y2 andy1y3. The reflection G with respect to γ exchanges y0 with y2 and y1 withy3.Now (F,G) ∈ I(H2)× I(H2) = H defines an orientation-reversing isometryφ of H2 ×H2 and, setting zi = (xi, yi), F induces the permutation (02)(13)of the zi. So, by H-invariance of f , we must have:

f(z0, z1, z2, z3) = −f(φ(z0), φ(z1), φ(z2), φ(z3)) =

= −f(z2, z3, z0, z1) = −f(z0, z1, z2, z3),

where the last equality is due to the fact that f is alternating.

Proof of Proposition 6.4.2. During the proof of Lemma 6.4.1, we have al-ready seen that, if xi and yi are two 5-tuples of cyclically ordered points inS1, the relation

Alt(π∗1Or ` π∗2Or)(

(x0, y0), (x1, y2), (x2, y4), (x3, y1), (x4, y3))

= 2/3

holds. But, whenever one takes away one pair of points, the remaining pairssatisfy the hypotheses of the previous lemma. Hence

(d3f)(

(x0, y0), (x1, y2), (x2, y4), (x3, y1), (x4, y3))

= 0

and(d3f+Alt(π∗1Or ` π∗2Or)

)((x0, y0), (x1, y2), (x2, y4), (x3, y1), (x4, y3)

)= 2/3.

130

6.5 Volume of straight simplices in H2 ×H2

It is interesting to notice that, unlike the hyperbolic case, the proportionalityconstant for H2 × H2 is not the inverse of the supremum of volumes ofstraight simplices. In particular, the inequality provided by Remark 5.1.14is not sharp for H2 ×H2. This is a consequence of the following, somewhatastonishing result, which will be the main subject of this section.

Proposition 6.5.1. For each ε > 0, there exists a straight 4-simplex inH2 ×H2 with volume greater than π2 − ε.

We remark that we do not know the exact upper bound to the volumesof straight simplices in H2 ×H2. However, a 4-simplex projects to (possiblydegenerate) geodesic pentagons in each H2-factor and is therefore containedin their product. Hence, a very coarse bound to the volumes of straight4-simplices is 9π2.We would like to thank Prof. Bucher-Karlsson for providing the main ideaof the proof of Proposition 6.5.1: we will show that, for each δ > 0, thereexists a straight 4-simplex in H2×H2 containing a product of subsets of H2,each with volume greater than π − δ.If R > 0, define p(R) as the point at hyperbolic distance R from the centreof the Poincaré disc, lying on the ray γ0 representing the ideal point (1, 0).If α ∈ [0, π), we can consider the ray γα starting at the origin and formingan angle α with γ0; let q(α,R) be the point on γα at a distance R from theorigin. We also define the triangle Tα,R := ∆(O, p(R), q(α,R)).If t ∈ (0, 1], we can consider for each x, y ∈ H2 the geodesic γt,x,y : [0, 1]→ H2

such that γt,x,y(0) = x and γt,x,y(t) = y. Given a geodesic triangle ∆(a, b, c)in H2 (the ordering of the vertices is important), we define the followingsubset of the hyperbolic plane:

Ht

(∆(a, b, c)

):=

⋃y∈∆(b,c)

γt,a,y([0, 1])

and the function fα,t : (0,+∞)→ (0,+∞) given by:

fα,t(R) := min R′ | Ht

(Tα,R

)⊆ Tα,R′ .

Notice that fα,t is well-defined since the set Ht(Tα,R) is compact.Now we can at last prove Proposition 6.5.1.

Proof of Proposition 6.5.1. Choose δ > 0 so that (π−δ)2 > π2−ε and choose0 < α < δ; since the supremum of the volumes of the Tα,R is π − α > π − δ,we can find R with vol(Tα,R) > π − δ.Let p′ 6= p(R) be a point on the geodesic segment from O to p(R) so thatvol(∆(O, p′, q(α,R))) > π − δ; let d be the hyperbolic distance between p′

131

and p(R) and set η := d/R.For each s ∈ (0, 1) consider the set:

Ωs :=⋃

y∈∆(O,p′)

γ1,q(α,R),y([0, s]) ;

clearly, we can choose s so that vol(∆(O, p′, q(α,R)) \ Ωs) > π − δ.Finally, set t := min(s, η) and define for each R > 0 the points

P (R) := p(fα,t(fα,t(R))) and Q(α,R) := q(α, fα,t(fα,t(R))) .

We will show that the straight 4-simplex in H2 ×H2:

Sα,R := ∆((O, q(α,R)

),(O, p(R)

),(O,O

),(P (R), O

),(Q(α,R), O

))contains the product

Tα,R ×(∆(O, p′, q(α,R)) \ Ωs

)and hence vol(Sα,R) > (π − δ)2 > π2 − ε.

Claim: the 3-simplex

∆((O, p(R)

),(O,O

),(P (R), O

),(Q(α,R), O

))contains the product Tα,fα,t(R) ×∆(O, p′).

Proof of the claim. Fix arbitrary points x ∈ Tα,fα,t(R) and y ∈ ∆(O, p′). Let

γ : [0, 1]→ H2 be the geodesic segment with γ(0) = p(R) and γ(1) = O. Lett ∈ (0, 1) be such that γ(t) = y; clearly t ≥ η ≥ t.We can find another geodesic segment γ′ : [0, 1] → H2 such that γ′(0) = O,its image γ′([0, 1]) is all contained in

∆(O,P (R), Q(α,R)

)= Tα,fα,t(fα,t(R))

and γ′(t) = x. Indeed:

Ht

(Tα,fα,t(R)

)⊆ Ht

(Tα,fα,t(R)

)⊆ Tα,fα,t(fα,t(R)) .

Then, by Lemma 6.1.5, a geodesic segment from the point (O, p(R)) to(γ′(1), O) in H2 × H2 is given precisely by (γ′, γ) and, at the time t, itpasses through (x, y). Since (γ′(1), O) lies in the simplex

∆((O,O

),(P (R), O

),(Q(α,R), O

))= ∆

(O,P (R), Q(α,R)

)× O ,

we obtain that (x, y) lies in ∆((O, p(R)), (O,O), (P (R), O), (Q(α,R), O)).

132

We will now conclude the proof by showing that the geodesic cone onTα,fα,t(R) ×∆(O, p′) with vertex (O, q(α,R)) contains the product

Tα,R ×(∆(O, p′, q(α,R)) \ Ωs

).

Take x ∈ Tα,R and y ∈ ∆(O, p′, q(α,R))\Ωs. We can find a geodesic segmentγ : [0, 1]→ H2 with γ(0) = q(α,R), with γ(1) lying on the segment ∆(O, p′)and γ(s) = y for some s ∈ [0, 1]; observe that it must be s > s ≥ t.Moreover, there exists another geodesic segment γ′ : [0, 1] → H2 such thatγ′(0) = O, its image γ′([0, 1]) is all contained in Tα,fα,t(R) and γ′(s) = x.Indeed:

Hs

(Tα,R

)⊆ Ht

(Tα,R

)⊆ Tα,fα,t(R) .

Again, we have obtained a geodesic segment (γ′, γ) from the point (O, q(α,R))to a point of Tα,fα,t(R) ×∆(O, p′) that passes through (x, y).

133

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