topological tameness of margulis spacetimeswmg/arxiv.org_pdf_1204.5308v2.pdf · topological...

TOPOLOGICAL TAMENESS OF MARGULISSPACETIMES

SUHYOUNG CHOI AND WILLIAM GOLDMAN

Dedicated to the memory of Bill Thurston, our mentor

Abstract. We show that Margulis spacetimes without parabolicholonomy are topologically tame. A Margulis spacetime is the quo-tient of the 3-dimensional Minkowski space by a free proper iso-metric action of the free group of rank ≥ 2. We will use our partic-ular point of view that the Margulis spacetime is a manifold-with-boundary with an RP3-structure in an essential way. The basictools are a bordification by a closed surface with an RP2-structureand a free holonomy group, the work of Goldman, Labourie, andMargulis on geodesics in the Margulis spacetimes and 3-manifoldtopology.

Contents

1. Introduction 22. Preliminaries 72.1. RPn-structures on manifolds 72.2. Projective S3 82.3. A matrix lemma 112.4. The space of properly convex sets and the joins 123. The oriented Lorentzian space and projective boosts 143.1. Oriented Lorentzian vector spaces 143.2. Null half-planes 143.3. Affine boosts 17

Date: June 10, 2013.1991 Mathematics Subject Classification. Primary 57M50; Secondary 53C15,

53C50, 53A20.Key words and phrases. Lorentzian manifolds, handlebody, geodesic flow, RP2-

structure, proper action.The first author was supported by the National Research Foundation of Ko-

rea(NRF) grant funded by the Korea government(MEST) (No.2010-0027001). Thesecond author gratefully acknowledges partial support from National Science Foun-dation grants DMS-070781 and DMS-1065965.

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2 CHOI AND GOLDMAN

3.3.1. Oriented axes 173.4. Projective boosts 184. RP2-surfaces and hyperbolic surfaces and their dynamics 194.1. RP2-structures on surfaces 194.2. Complete hyperbolic surfaces and Fuchsian groups 204.3. The dynamic of geodesic flows on hyperbolic surfaces 215. RP2-Schottky uniformizations 235.1. The construction of the Γ-invariant domain 235.2. The RP2-surface 266. The dynamic core of a Margulis spacetime 296.1. Margulis invariants 296.2. Neutralized sections 306.3. Lifting the neutralized section to the coverings 326.4. Mapping the convex core 327. Γ acts properly on E ∪ Σ. 337.1. The action of projective-boost automorphisms 347.2. The proper discontinuity of the action of Γ. 378. The proof of tameness. 469. Asymptotics of Margulis spacetimes 499.1. Supporting hyperplanes 499.2. Domains 5210. Geometric finiteness 55References 58

1. Introduction

A complete affine manifold is a quotient M of an n-dimensionalreal affine space E by a discrete group Γ of affine transformations act-ing properly. Equivalently, M is a manifold with a geodesically flattorsionfree affine connection. The purpose of this paper is topologi-cally classify complete affine manifolds in dimension n = 3 when thefundamental group π1(M) is finitely generated and has no parabolicholonomy.

Those affine 3-manifolds where the fundamental groups are solvablewere classified by Fried-Goldman [29] in the twentieth century. Theseinclude the case when the manifolds are compact.

In all other cases, M admits a parallel Lorentzian structure, andπ1(M3) ∼= Γ is a free group. The existence of these manifolds was firstdemonstrated by Margulis [41], and later clarifed by Drumm [22] usingthe geometric construction of crooked planes. In particular Drumm’s

MARGULIS SPACETIMES 3

examples are homeomorphic to the interiors of closed solid handlebod-ies, that is, compact 3-manifolds-with-boundary obtained by attaching1-handles to a closed 3-ball. We call such a manifold M topologicallytame.

Charette, Drumm, Goldman, and Labourie [22], [23], [24] made con-tributions to understanding such spacetimes with parallel Lorentzianstructures culminating in recent work [35] and [34]. The completeclassification of such spacetimes is continuing with work of Charette,Drumm, Goldman, Margulis and Minsky [12], [13], [36].

The theory of compactifying open manifolds goes back to Browder,Levine and Livesay [10] and Siebenmann [47]. For 3-manifolds, Tucker[50] and Scott and Tucker [49] made an initial clarification. See alsoMeyers [44]. As a note, we state that complete hyperbolic 3-manifoldswith finitely generated fundamental groups were shown to be tame byAgol, Calegari, and Gabai. See Bowditch [9] for details on this verylarge topic. We won’t elaborate on this as the methods are completelydifferent. Our proof is closer to the proof of the tameness of geomet-rically finite hyperbolic 3-manifolds due to Thurston [48]. (This iswritten up nicely by Epstein and Marden [26] and Marden [42] evenearlier. See also Bonahon [7].)

Lately after the first version of this paper was uploaded, there isan another independent approach to this question by J. Danciger, F.Gueritaud, and F. Kassel. Their approach uses the deformation ofconstant curvature Lorentzian manifolds and yields other results suchas the deformability of Margulis spacetimes to anti-de Sitter spaces.Compared to their approach, our approach mostly concentrates on RP2-structures and the compactification of the spacetime. We also obtainsome asymptotic properties of the action.

Let V denote a 3-dimensional real vector space with an inner productof signature 2, 1 and a fixed orientation. The spacetime E is an affinespace with underlying vector space V. A Lorentzian isometry is anautomorphism of E preserving the Lorentzian inner product. Denotethe group of orientation-preserving Lorentzian isometries by Isom+(E).The projectivization P(V) is defined as the quotient space

V − O/ ∼ where v ∼ w if and only if v = sw for s ∈ R− 0.

Of course, P(V) is identical with the real projective 2-space RP2. Recallthat we embed the hyperboloid model H2 of the hyperbolic plane to thedisk that is the interior of the conic in RP2 determined by null vectors.Here PGL(3,R) acts faithfully on RP2 as the group of collineations.

4 CHOI AND GOLDMAN

Define S(V) as

V − O/ ∼ where v ∼ w if and only if v = sw for s > 0.

Then the image S+ of the space of future timelike vectors identifieswith the hyperbolic 2-plane H2, which is basically the Beltrami-Kleinmodel of the hyperbolic plane. Let S− denote the subspace of S(V)corresponding to past timelike vectors. The group of orientation pre-serving linear maps of V is denoted by SO(2, 1), a Lie group with twocomponents one of which is the identity component SO(2, 1)o. Thelinear group SO(2, 1)o acts faithfully on H2 = S+ as the orientation-preserving isometry group and SO(2, 1) acts so on S+ ∪S− and acts onS(V) projectively where the action is induced from that on V.

Since SO(2, 1) injects to PGL(3,R) under projection GL(3,R) →PGL(3,R), we consider that SO(2, 1) acts on S+ = H2 a subset of RP2

as well in a projective manner. Note that SO(2, 1)o acts on S+ honestlyas a subset of S(V).

Recall that there is a homomorphism

L : Isom+(E)→ SO(2, 1)

given by taking the linear part acting on S(V). We also denote by

L′ : Isom+(E)L→ SO(2, 1) → PGL(3,R)

with the image acting on RP2.A Lorentzian transformation is said to be parabolic if its linear part

has only eigenvalues 1 but is not identity. One that is neither para-bolic nor identity has eigenvalues λ, 1/λ, 1. Such an element is calleda hyperbolic transformation if |λ| > 1. A positive hyperbolic transfor-mation is a hyperbolic transformation with only positive eigenvalues.Sometimes a positive hyperbolic transformation is called a linear boost.

Suppose that Γ is a finitely generated Lorentzian isometry groupacting freely and properly on E. We assume that Γ is not amenable(i.e., not solvable). Since E/Γ is aspherical, the fundamental group Γis torsion-free. Γ injects under L to L(Γ) ⊂ SO(2, 1) acting properlydiscontinuously and freely on S+ ∪ S−. By Mess [43], Γ is a free groupof rank ≥ 2. (See [29] where this nontrivial fact was first discoveredand Section 3 of [34] for details.) Thus, we restrict ourselves to thegroup actions of free groups of rank ≥ 2 in this paper. Then E/Γ issaid to be a Margulis spacetime, and an element of Γ is said to be aholonomy.

Note that H2/L′(Γ) may be not orientable if L(Γ) is not a subset ofSO(2, 1)o. However, S+ ∪ S−/L(Γ) is an orientable surface double-covering H2/L′(Γ). (See [13] for details.) The hyperbolic surface


H2/L′(Γ) is geometrically finite if Γ is nonelementary and has no parabol-ics.

In this paper we will be concerned with the cases without parabolicelements in Γ. The following conditions are equivalent conditions for afinitely generated Lorentzian isometry group Γ.

• E/Γ is a Margulis spacetime without parabolics holonomies.• Γ is proper affine deformation of a geometrically finite Fuchsian

group; i.e., H2/L′(Γ) is geometrically finite and Γ acts properlyon E. (See [35] for definition. )• Γ is a nonamenable Lorentzian isometry group acting properly

on E without parabolic holonomy.• Γ is a free Lorentzian isometry group of rank ≥ 2 acting prop-

erly and freely on E without parabolic holonomy.

In this case, we say that Γ is a Margulis group without parabolics.It also follows from above that elements of Γ are either hyperbolic

or parabolic or identity. (See [35]). Notice that the definitions ofhyperbolic, positive hyperbolic, and parabolic for elements of Γ are thesame as ones for L′(Γ) as a group of isometries of S+. (If a hyperbolicelement in SO(2, 1) has a negative eigenvalue, then they should be−λ,−1/λ, 1, 1.)

An RPn-structure on a manifold is given by an atlas of charts to RPnwith projective transition maps. Such geometric structures were firstconsidered by Kuiper, Koszul, Benzecri and others in the 1950s and1960s. Further developments can be followed in Goldman [33], Choiand Goldman [18], and Cooper, Long, and Thistlethwaite [19] and [20].

Theorem 1.1. Let Γ be a finitely generated free group of rank ≥ 2in SO(2, 1) acting on S+ ∪ S− properly discontinuously and freely butwithout parabolics. Let S := S(V) be the sphere of directions in V,

which is a model for the universal covering space RP2 −→ RP2. Thenthere exists a Γ-invariant open domain D ⊂ S(V) such that D/Γ is aclosed surface Σ. Such a domain is unique up to the antipodal map A.

These surfaces have the RP2-structures on closed surfaces of genusg, g ≥ 2, discovered by Goldman [31] in the late 1970s. Althoughtheir developing maps from the universal cover Σ of some such surfaceΣ are not covering-spaces onto their images in RP2, when lifted to

the double-covering space S(V) := RP2 −→ RP2, the developing maps(remarkably) are covering-spaces onto open domains. The surface isa quotient of a domain in S by a group of projective automorphisms,an RP2-analog of the standard Schottky uniformization of a Riemannsurface as a CP1-manifold as observed by Goldman.

6 CHOI AND GOLDMAN

An n-dimensional open manifold is said to be tame if it is home-omorphic to the interior of a compact n-manifold with boundary. Ahandlebody is a 3-dimensional manifold obtained from a 3-ball B3 byattaching 1-handles to 3-dimensional balls. The topology of handle-bodies and related objects form a central topic in the 3-dimensionalmanifold topology. See Hempel [38] for a thorough survey of this fieldof mathematics developed in the 20th century.

Theorem 1.2. Let E be the standard Lorentzian space with the stan-dard norm given by x1

0−x21−x2

2, and let Γ be a Margulis group withoutparabolics. Then E/Γ is homeomorphic to the interior of a handlebodyof genus r for the rank r of Γ as a free group.

We define in this paper that a complete geodesic on a quotient spaceof a hyperbolic space or a Lorentzian space is nonwandering if it isbounded in both directions; that is, the closure of the forward part iscompact and so is that of the backward part; i.e., the α-limit and theω-limit are both nonempty and compact. This is the same as the term“recurrent” in [34] and [14], which does not agree with the commonusage in the dynamical system theory whereas the recurrence set theydiscuss is the same as the nonwandering set as in Eberlein [25] andKatok and Hasselblatt [39]: the reason is that these are geodesics withboth endpoints in the limit sets and the set of these geodesics coincidewith the nonwandering set in [25] by Corollary 3.8 there.

The work of Goldman–Labourie–Margulis [35] on nonwanderinggeodesics on M implies tameness: A Margulis spacetime is geometri-cally finite in some sense since every closed geodesic is in a boundedneighborhood, which will explain our particular viewpoints on this inSection 9.

In Section 2, we review facts on hyperbolic surfaces, Fuchsian groupactions, RPn-structures and geometry that we need. We compactify Eby a real projective 2-sphere S that is just the space of directions in E.

In Section 3, we review facts about the Lorentzian space E and affineboosts. The affine boosts will be extended to projective boosts in S3.

In Section 4, we review real projective structures on surfaces andhyperbolic surfaces. We also discuss the dynamic of the geodesic flowson the unit tangent bundles of hyperbolic surfaces.

In Section 5, we prove Theorem 1.1. This will be done by finding anopen domain Σ in the boundary S of E where Γ acts properly discon-tinuously and freely. Σ/Γ is shown to be a closed RP2-surface Σ. Sucha surface was constructed by Goldman [31]; however, we realize it asthe quotient of an open domain in the sphere S, which is a union oftwo disks and infinitely many strips joining the two.


Now let Γ be as assumed in Theorem 1.2. In Section 6, we recallthe work of Goldman, Labourie, and Margulis [35] on nonwanderinggeodesics on the Margulis spacetime. Their work in fact shows thatall nonwandering spacelike geodesics in a Margulis spacetime are in auniformly bounded part.

In Section 7, we prove the proper discontinuity of the action of thegroup Γ on E ∪ Σ. We first show that for a fixed Lorentzian isom-etry g, we can push every compact convex subset of E ∪ Σ into anε-neighborhood of the “attracting” segment of g in S by gi for suffi-ciently large i. (This is the key model of our proof.)

We next show that for any compact set K ⊂ E ∪ Σ, there are onlyfinitely many g ∈ Γ so that g(K) ∩ K 6= ∅. Suppose not. We find asequence gi so that gi(K)∩K 6= ∅. Using the work [35], we can findan infinite sequence gi that behaves almost like gi in the dynamicalsense as i→∞ up to small changes of stable and unstable planes. Inother words, we can find a uniformly bounded sequence of coordinatesto normalize gi into fixed forms and these coordinates are uniformlyconvergent.

In Section 8, we prove the tameness of E/Γ, i.e., Theorem 1.2, usingthe classical 3-manifold topology as developed by Dehn and so on.

In Section 9, we show that there exists an invariant set of hyper-planes asymptotic to the limit set Λ of Γ on Cl(S+). This gives us twodisjoint Γ-invariant closed convex domains in E asymptotic to Λ andits antipodal set Λ− in S. (This is a generalization of a result of Mess[43].)

In Section 10, we also define “geometric finiteness” in this settingand show that there exists a topological core of a Margulis spacetimethat contains all spacelike nonwandering geodesics.

This work was started from the question of Goldman on the RP2-bordification of the Margulis spacetime during the Newton InstituteWorkshop “Representations of Surface Groups and Higgs Bundles” heldin Oxford, March 14-18, 2011. The authors thank the Newton Instituteand the Institut Henri Poincare where parts of this work were carriedout. Finally, we thank Bill Thurston without whose teaching we couldnot have accomplished many of the things in this paper.

2. Preliminaries

2.1. RPn-structures on manifolds. The projective space RPn is givenby P(Rn+1). The general linear group is sent

GL(n+ 1,R)→ PGL(n+ 1,R) := GL(n+ 1,R)/ ∼

8 CHOI AND GOLDMAN

where two linear maps L1 and L2 are equivalent if L1 = sL2 for s ∈R− 0. A map of an open subset of RPn to one of RPn is projectiveif it is a restriction of an element of PGL(n+ 1,R).

More generally, a nonzero linear map L : Rn+1 → Rn+1 becomes aprojective endomorphism

L : P(Rn+1)− P(N)→ P(Rn+1)

where N is the kernel of L and P(N) is the image of N−O in P(Rn+1)

defined by L([~v]) = [L(~v)]. (See Section 5.1 of Benzecri [6].) The spaceof projective endomorphisms equals the projectivization P(Matn+1(R))of the linear space Matn+1(R) of all linear maps Rn+1 → Rn+1. Thespace P(Matn+1(R)) forms a compactification of the group PGL(n +1,R) as observed first by Kuiper [40] and developed by Benzecri [6].

The projective geometry is given by a pair (RPn,PGL(n + 1,R)).An RPn-structure on a manifold M is given by a maximal atlas ofcharts to RPn with projective transition maps. The manifold M withan RPn-structure is said to be an RPn-manifold. A projective map fortwo RPn-manifolds M and N is a map f : M → N so that φ f ψ−1

is projective whenever it is defined where φ is a chart for N and ψ−1

is a local inverse of a chart for M .An RPn-structure on a manifold M gives us an immersion dev : M →

RPn that is equivariant with respect to a homomorphism

h : π1(M)→ PGL(n+ 1,R) : that is,

dev γ = h(γ) dev for every γ ∈ π1(M).

Here, dev is called a developing map and h is called the holonomyhomomorphism. (dev, h) is only determined up to an action of PGL(n+1,R) where

g(dev, h(·)) = (g dev, gh(·)g−1) for g ∈ PGL(n+ 1,R).

Conversely, such a pair (dev, h) will give us an RPn-structure sincedev gives us charts that have projective transition maps. (See [16] fordetails.)

2.2. Projective S3. We can identify Sn with S(Rn+1). There exists aquotient map q1 : Sn → RPn given by sending a unit vector to its equiv-alence class. This is a double-covering map, and it induces an RPn-structure on Sn. The group Aut(Sn) of projective automorphisms is iso-morphic to the group SL±(n+1,R) of linear maps of determinant equalto ±1 with the quotient homomorphism SL±(n+1,R)→ PGL(n+1,R)with the kernel ±I. It will be convenient to think of these as matricesin SL±(n+ 1,R).


This space is again an open dense subspace of S(Matn(R)). We callthe element of S(Matn(R)) a projective endormorphism of Sn. Each

element corresponds to a map L : S(Rn+1) − S(N) → S(Rn+1) whereN is the kernel of a linear map L in Matn(R) and S(N) is its image inSn and defined by

L([~v]) = [L(~v)]

for ~v ∈ Rn+1.For an open domain D in Sn and a discrete group Γ in Aut(Sn) acting

on it properly discontinuously and freely, D/Γ has an RPn-structuresince q1|D gives an immersion and the homomorphism

Γ ⊂ Aut(Sn)→ PGL(n+ 1,R)

gives the associated holonomy homomorphism.In this paper, we will study objects on S3 = S(R4) = S(R⊕ R3). A

homogeneous coordinate system of S3 is given by setting a point p of

S3 be given coordinates [x : y : z : t] where (x,y,z,t)|(x,y,z,t)| represents p as a

unit vector in S3. Again

[x : y : z : t] ∼ [x′ : y′ : z′ : t′] iff (x, y, z, t) = λ(x′, y′, z′, t′) for λ > 0.

There is the antipodal map

A : S3 → S3 given by [v]→ [−v].

Given a subset or a point K of S3, we denote by K− the set of antipodalpoints of points of K.

A subspace of S3 is a subset defined by a system of linear equations inR4. A line is a 1-dimensional subspace. A singleton is not a subspace.A projective geodesic is an arc in a line. We will be using the standardRiemannian metric on S3 to be denoted by dS3 . Notice that the geo-desic structure of S3 is the same as the projective one. Thus, we willuse the same term “geodesics” for both types of geodesics. Geodesicsand totally geodesic subspaces are all subsets of lines or subspaces inS3. A projective automorphism sends these to themselves, while it doesnot preserve any metric. A pair of antipodal points on S3 is the equiv-alence class [v] of a nonzero vector v and one [−v] of −v. A segmentconnecting two antipodal points is precisely a segment of dS3-length π.

Here, we denote by pqp− the unique closed segment in S3 with end-points p and p− passing through q not equal to p or p−. By pq, wedenote the unique closed segment in S3 with endpoints p and q notcontaining p−, q− provided p 6= q and p 6= q−. The notation pqo orpqp−

o indicates the interior of the segment.

10 CHOI AND GOLDMAN

The Lorentzian space E is the affine space R3 equipped with a non-degenerate bilinear form B of signature 1, 1,−1. If we choose the originin E, we obtain the Lorentzian vector space V.

One can think of R3 to be identical with E and the complement ofa subspace of codimension-one in RP3, so called the complete affinesubspace. R3 = E lifts to a subspace of S3 double-covering RP3. Theclosure of the lifted E is a standard 3-dimensional hemisphere H , andE identifies with the open hemisphere H o. The boundary S2 of E is thesubspace of codimension-one identifiable with the projective 2-sphereS of directions in R3. (We identify E with the open affine space givenby t > 0 in S3 from now on for convenience.)

The sphere S corresponds to the hyperplane given by t = 0 and theorigin of E is given by [0 : 0 : 0 : 1], and by choosing O. For this systemof coordinates, the point (x, y, z) ∈ E is given coordinates [x : y : z : 1].

S also has a homogeneous coordinate system [x, y, z] assigned to a

vector p in S of unit length 1 if (x,y,z)||(x,y,z)|| = p. In the larger coordinate

system [x : y : z] is identical with [x : y : z : 0].To summarize, the subsets of S3 indentified in this paper are:

S3 = [x : y : z : t]|x, y, z, t ∈ R, (x, y, z, t) 6= (0, 0, 0, 0)H = [x : y : z : t]|x, y, z, t ∈ R, t ≥ 0, (x, y, z, t) 6= (0, 0, 0, 0)R3 = E = H o = [x, y, z, t]|x, y, z, t > 0 ∈ R

= [x : y : z : 1]|x, y, z ∈ RS = [x : y : z : 0]|x, y, z ∈ R, (x, y, z, t) 6= (0, 0, 0, 0).

We will also be using a fixed Euclidean metric dE on E compatiblewith the affine coordinate system. It is of great importance that pro-jective geodesics, spherical geodesics, and Euclidean geodesics on E arethe same ones up to parametrizations.

The group Isom+(E) of orientation-preserving Lorentzian transfor-mations of E is in the group Aff+(E) of orientation-preserving affinetransformations of E. Aff+(E) identifies with the subgroup of Aut(S3)of orientation-preserving elements acting on R3 = E, preserving S, sinceany affine transformation of R3 extends to a projective automorphismof RP3 and hence an automorphism of S3 analytically. Often such atransformation is represented by

(A ~v0 1

)∈ GL(3,R)


a representative matrix in Aut(S3) with A a nonsingular 2 × 2-matrixin SO(2, 1) and requiring the last diagonal diagonal entry 1. It is rep-resented as an affine transformation

~x 7→ A~x+ ~v for ~x ∈ E.

The elements of respective groups are called projective automor-phisms of Lorentzian type and ones of affine type or just Lorentzianisometries or affine transformations. A bi-Lipschitz map of S3 is ahomeomorphism f : S3 → S3 so that

C−1dS3(x, y) ≤ dS3(f(x), f(y)) ≤ CdS3(x, y) for x, y ∈ S3

where C > 0 is independent of x, y. Elements of Aut(S3) including onesextending the Lorentzian isometries are bi-Lipschitz maps of S3.

Finally, for the purpose of drawing figures only, we map E to the unit3-ball in R3 by the map

[x : y : z : 1]→ (x, y, z)√x2 + y2 + z2 + 1

.

This will identify S with the sphere of radius 1 in R3 and as a codimension-one totally geodesic subspace of S3, represented stereographically.

We will denote by bdA or bdXA the topological boundary of a subsetA of some topological space X and by intA or intXA the topologicalinterior. If A is a topolgical manifold with boundary, we will denotethe manifold boundary by ∂A and we will denote the manifold interiorby Ao.

2.3. A matrix lemma. We denote the standard vector by

~e1 = (1, 0, 0, 0), ~e2 = (0, 1, 0, 0), ~e3 = (0, 0, 1, 0), ~e4 = (0, 0, 0, 1),

and denote

e1 = [~e1], e2 = [~e2], e3 = [~e3], e4 = [~e4],

e1− = [−~e1], e2− = [−~e2], e3− = [−~e3], e4− = [−~e4].

Lemma 2.1 (Uniform convergence). Let vji for j = 1, 2, 3, 4 be a se-

quence of points of S3. Suppose that vji → vj∞ for each j and mutuallydistinct points v1

∞, . . . , v4∞ formed by independent vectors in R4. Then

we can choose a sequence hi of elements of Aut(S3) so that

• hi(vji ) = ej,• hi and h−1

i are represented by uniformly bounded matrices,• the sequence hi converges to h∞ uniformly for h∞ and• h−1

i converges to h−1∞ uniformly both under Cs-topology for

every s ≥ 0.

12 CHOI AND GOLDMAN

Proof. Let ~vji represent the unit norm vector corresponding to vji . We

find hi ∈ Aut(S3) so that hi(ej) = vji for each i. We set Mi(~ei) = ~vji ;that is, the column vectors of Mi equals ~v1

i , ~v2i , ~v

3i , ~v

4i . Then clearly, the

sequence Mi converges to a 4×4-matrix M∞ of nonzero determinant.Therefore, the sequence M−1

i of inverse matrices converges to M−1∞ .

We let hi be the projective automorphism in Aut(S3) induced byM−1

i , let h∞ be the one corresponding to M−1∞ , and this satisfies the

properties we need.

2.4. The space of properly convex sets and the joins. LetdS3-diam(A) denote the diameter of the set, that is, the supremum ofthe set of distances between every pair of points of a subset A of Sn.Given a pair of points or subsets A and B in Sn, we define the infimumdistance

dS3(A,B) = infdS3(x, y)|x ∈ A, y ∈ B,

and the supremum distance

dsS3(A,B) = supdS3(x, y)|x ∈ A, y ∈ B = dS3-diam(A ∪B).

If dS3(A,B) > ε for ε > 0, then we say that A is bounded away from Bby ε or conversely.

An ε-neighborhood Nε(A) for a number ε > 0 of a subset A of S3

is the set of points of dS3-distances less than ε from some points ofA. We define the geometric Hausdorff distance dHS3(A,B) between twocompact subsets A and B of S3 to be

infε ≥ 0|B ⊂ Nε(A) and A ⊂ Nε(B).

A sequence of compact sets Ki in S3 converges or converges geomet-rically to a compact subset K if for every ε > 0, there exists N so thatfor i > N , dHS3(K,Ki) < ε or equivalently

K ⊂ Nε(Ki) and Ki ⊂ Nε(K).

We will simply write Ki → K for Ki and K compact subsets of S3.The following are commonly known facts with elementary proofs.

Proposition 2.2. • The space of compact subsets of a compactHausdorff space with this metric is compact Hausdorff also andhence every sequence has a convergent subsequence.• given a convergent sequence Ki of compact subsets of S3, if a

sequence Ji is such that Ji ⊂ Ki, then any geometric limit ofa subsequence of Ji is a subset of the geometric limit of Ki.


• If we have Kji → Kj for each j = 1, . . . ,m, then

n⋃j=1

Kji →

n⋃j=1

Kj

holds.

Note that

dS3(A,B) ≤ dHS3(A,B) ≤ dsS3(A,B),

and three could be all distinct.Suppose that two compact subsets A and B of S3 have no pairs of

points x ∈ A, y ∈ B so that x and y are antipodal, i.e., A ∩A(B) = ∅.Then the join J(A,B) is the union of all segments of dS3-lengths < πwith single endpoints in A and the other ones in B. In set theoreticterms, we let

(1) J(A,B) = q1(c1~u1 + c2~u2|c1, c2 ≥ 0, ~u1 ∈ q−11 (A), ~u2 ∈ q−1

1 (B)).

When A is a great circle in S3 and B is a point 6∈ A, every segmentfrom B to A is of dS3-length < π − ε for a fixed ε > 0. In this case,J(A,B) is a 2-dimensional hemisphere, a totally geodesic disk withgeodesic boundary in S3, a convex set.

If A and B are compact and has no antipodal pair of a point of Aand a point B, then J(A,B) is well-defined and is also compact as wecan easily show that J(A,B) is closed.

If A and B are convex and A ∪ B has no pair of antipodal points,then J(A,B) is defined and also properly convex. The proof followsby reverting back to R4 by (1). Also, for a projective automorphismg ∈ Aut(S3), we obtain

gJ(A,B) = J(g(A), g(B)).

Lemma 2.3. Let Ai and Bi be a sequence of properly convex compactsubsets of S3 geometrically converging to properly convex compact setsA and B respectively. Suppose that A∩A(B) = ∅. Then Ji = J(Ai, Bi)is defined for i > I0 for sufficiently large I0 and the sequence Jii>I0converges to the join J of A and B.

Proof. Clearly, Ai and Bi have no antipodal pair of points for suffi-ciently large i as Ai ⊂ Nε(A), Bi ⊂ Nε(B) for sufficiently large i. Onecan parameterize each geodesic segment connecting a point of Ai tothat of Bi by arclength in dS3 . From this, the lemma follows.

14 CHOI AND GOLDMAN

3. The oriented Lorentzian space and projective boosts

3.1. Oriented Lorentzian vector spaces. Let (V, ·,Det) denote athree-dimensional oriented Lorentzian R-vector space. That is, V ∼= R3

and is given a symmetric bilinear form

B : V × V −→ Rdefined by (v, u) 7→ v · u of index 1 and a nondegenerate alternatingtrilinear form

V × V × V −→ Rdefined by (v, u,w) 7→ Det(v, u,w). Its automorphism group is the spe-cial orthogonal group SO(2, 1).

The Lorentzian structure divides S into three open domains S+,S0,and S− separated by two conics bdSS+ and bdSS−. Let S+ ⊂ S de-note the set of future timelike directions; its boundary bdSS+ consistsof future null directions. Similarly, let S− ⊂ S denote the set of pasttimelike directions with boundary bdSS− consisting of past null direc-tions. Both S+ and S− are cells, interchanged by A . The set S0 ofspacelike directions forms an A -invariant annulus bounded by the dis-joint union bdSS+

∐bdSS−.

Recall that S+ is the Beltrami-Klein model of the hyperbolic planewhere SO(2, 1)o acts as the orientation-preserving isometry group. Herethe metric geodesics are precisely the projective geodesics and viceversa. (We will use the same term “geodesics” for both types ofgeodesics for ones in S+ and later S−.) The geodesics in S+ are straightarcs and bdSS+ forms the ideal boundary of S+. For a finitely gener-ated discrete subgroup Γ in SO(2, 1)o, S+/Γ has a complete hyperbolicstructure as well as an RP2-structure with the compatible geodesicstructure.

Alternatively, we can use S− as the model and S−/Γ has a completehyperbolic structure as well as an RP2-structure. Since S+ and S− areproperly convex, these are examples of properly convex RP2-surfaces.

3.2. Null half-planes. Let N denote the nullcone in V, that is, thecollection of all vectors v ∈ V with v · v = 0. Its projectivizationP(N − O) consists of all null lines in V. If v ∈ N − O, then itsorthogonal complement v⊥ is a null plane which contains the line Rv.The line Rv separates v⊥ into two half-planes.

The orientation on V determines an SO(2, 1)-invariant way to uniquelyassociate a component of v⊥−Rv to Rv as follows. Since v ∈ N holds,its direction [v] lies in either bdSS+ or bdSS−. Choose an arbitrary ele-ment u of S+ or S− respectively, so that the directions of v and u bothlie in Cl(S+) or Cl(S−) respectively. (For example u = (0, 0,±1) would


be sufficient.) Define the null half-plane W (v) (or the wing) associatedto v as:

W (v) := w ∈ v⊥ | Det(v,w, u) > 0.Since the Lorentzian product u × v is in W (v), the directions in thewing satisfy the right-hand rule. Since W (v) = W

(A (v)

)holds, the

null half-plane W (v) depends only on Rv. The corresponding set ofdirections is the open arc

ε(v) := [W (v)]

in S0 joining [v] to A [v] = [−v]. Since v⊥ is tangent to N , the arcε(v) is tangent to bdSS+. The orientation of bdSS+ as the boundaryorientation of S+ agrees with the orientation of ε(v) away from [v].

The corresponding map [v] 7−→ ε(v) is a SO(2, 1)-equivariant map

bdSS+ → Swhere S denotes the set of half-arcs of form ε(v) for v ∈ bdSS+. Thearcs ε([v]) for v ∈ bdSS+ foliate S0 obtained from an S1-action fixing[±1 : 0 : 0] where S1 is a subgroup of SO(2, 1). Let us call the foliationF .

Hence S0 has a SO(2, 1)-equivariant quotient map

Π : S0 → P(N − O) ∼= S1

and ε([v]) = Π−1([v]) for each v ∈ N − O.Consider a future-pointing null vector of Euclidean length

√2:

nθ :=

cos(θ)sin(θ)

1

for θ ∈ [0, 2π).

Then W (nθ) consists of all

pθ(t, s) := tnθ + s

− sin(θ)cos(θ)

0

where t ∈ R and s > 0. The arc cθ := ε(nθ) on the sphere of directionsS is parametrized by unit vectors

1√2

pθ(t,√

1− t2)

as −1 < t < 1. (As an element of S3, its homogeneous coordinatesequal

[1√2

pθ(t,√

1− t2) : 0]

where 1√2pθ(t,

√1− t2) is used as the last three coordinates.)

16 CHOI AND GOLDMAN

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

Figure 1. Thetangent geodesicsto disks S+ and S−in the unit sphereS imbedded in R3.

-4 -2 0 2 4

-4

-2

0

2

4

Figure 2. Thetangent geodesicsto disks S+

and S− in thestereographicallyprojected S from(0, 0,−1). Radialarcs are geodesics.The inner cir-cle representsthe boundary ofS+. The arcsof form ε(x) forx ∈ bdSS+ areleaves of the fo-liation F on S0.

The other component of v⊥−Rv could also be used as a wing W (v),but for the opposite orientation on V. Alternatively, these negativelyoriented null half-planes are the images of the positively oriented nullhalf-planes under the antipodal map A (which reverses orientation indimension three). This phenomenon appears in the theory of crookedplanes, where one must choose a class of orientations for crooked planes,as well as the Margulis invariant, where the choice of orientation playsa crucial role.

Since we have no need to consider negatively oriented null half-planes, we henceforth restrict our attention to positively oriented nullhalf-planes in this paper.


3.3. Affine boosts.

Definition 3.1. An isometry of E is an (affine) boost if its linear partis a positive hyperbolic element of SO(2, 1) and it acts freely on E.

Suppose that γ ∈ Isom+(E) is an affine boost. Then γ preservesa unique line which we denote Axis(γ). (Margulis calls this line Cγ.)Furthermore Axis(γ) is spacelike and the restriction of γ to Axis(γ) isa nonzero translation. In a suitable coordinate system, we can takeAxis(γ) to be the z-axis, in which case γ is the affine transformation:

pγ7−→

el 0 00 1 00 0 e−l

p+

0α0

where el 0 0

0 1 00 0 e−l

is the linear part L(γ) of γ.

3.3.1. Oriented axes. As observed by Margulis, the axis of an affineboost γ admits a canonical orientation, induced by the orientation ofE.

Let g = L(γ) ∈ SO(2, 1) be positive hyperbolic. Then since g isan isometry, and unimodular, the eigenvalues of g are λ, 1, λ−1 whereλ > 1. The eigenspaces for eigenvalues λ, λ−1 are null (since g is anisometry); choose respective eigenvectors v+(g), v−(g) ∈ V with thesame causal character (that is, they are either both future-pointingor both past-pointing). The 1-eigenspace Fix(g) ⊂ V is spacelike andorthogonal to v+(g) and v−(g).

Definition 3.2. The neutral eigenvector of g is the unique unit-spacelikevector v0(g) such that v+(g), v0(g), v−(g) is a positively oriented basisof V. (See [24] for details.)

Since Axis(γ) is parallel to Fix(L(g)

), the neutral eigenvector L(g)0

defines an orientation on Axis(γ). Denote the corresponding oriented

geodesic by Axis(γ).The weak-stable plane Wwu(γ) ⊂ E is the affine subspace containing

Axis(γ) and parallel to v−(g). Note that γn|Wwu(γ) acts on a point pof Wwu(γ) so that

dS3(gn(p), [v−(g) : 0])→ 0 and dS3(gn(p),Axis(γ))→ 0 as n→∞.(Notice Wwu(γ) is a null-space.)

18 CHOI AND GOLDMAN

3.4. Projective boosts.

Definition 3.3. A projective boost is a collineation of S3 defined bythe direct sum of a positive hyperbolic element of SL(3,R):[

el 00 e−l

]and a nontrivial unipotent element of SL(3,R):[

1 α0 1

],

where α 6= 0, such that the three fixed points lie on the ideal plane S∞.

Thus a projective boost is defined by the 4 × 4 matrix where theideal plane subspace is defined by the fourth coordinate:

el 0 0 00 1 0 α0 0 e−l 00 0 0 1

The three fixed points on P∞ are:

x+ := [1 : 0 : 0 : 0], x0 := [0 : 1 : 0 : 0], x− := [0 : 0 : 1 : 0]

in homogeneous coordinates.Such a collineation induces the affine isometry of E defined by:

vγ7−→

el 0 00 1 00 0 e−l

v +

0α0

The subspace invariant under the unipotent direct summand defines aprojective line, which we call the spine:

σ := [0 : ∗ : 0 : ∗]that contains the axis.

Thus, any conjugate element in Aut(S3) is also said to be a projectiveboost. We denote by x+(γ), x0(γ), x−(γ), and σ(γ) the points going tox+, x0, x− and the subspace σ under the coordinate change of γ to makeits matrix into the above form. The span W wu(γ) of x−(γ)∪σ(γ) is thesubspace of S3 extending the weak-stable plane Wwu(γ) ⊂ E. Pointson Wwu(γ) ⊂ E has x0(γ) as a limit point along the line σ(γ)\x0(γ).(Here σ(γ) is the subspace spanned by Axis(γ) in E.)

Let γn be a sequence of projective boosts in Aut(S3). Since S(Mat4(R))is compact, we can extract a convergent subsequence always. Supposethat limn→+∞ γ

n is a projective endormorphism γ∞ in S(Mat4(R)) of


rank one in S(Mat4(R)). (One could have a rank 2 limit also in gen-eral.) Then the undefined hyperplane of γ∞ equals a subspace W wu

∞of codimension-one and whose image equals a pair of points x+

∞ andx+∞−where

W wu(γn)→ W wu∞ , x+(γn)→ x+

∞.

More generally, if γn is a sequence in Aut(S3) converging to aprojective endomorphism γ∞ in S(Mat4(R)) of rank 1, we can choosea subsequence of γn so that the corresponding subsequence of theattracting fixed points x+(γn) converges to the image I(γ∞) andthe corresponding subsequence of the extended weak-stable subspacesW wu(γn) converges to the undefined subspace U(γ∞). (Note that wecan have a limit of rank 2 if we do not have v(γi)/λ(γi) → 0.)

4. RP2-surfaces and hyperbolic surfaces and theirdynamics

4.1. RP2-structures on surfaces. Let ϑ correspond to an element Aof SL(3,R) diagonalizable with three distinct positive eigenvalues. Letϑ′ be the element of Aut(S2) induced by A. Then ϑ′ has six fixed pointsa, a−, r, r−, s, s− and three great circles l1, l2, and l3 so that

a, r, a−, r− ⊂ l1, r, s, r−, s− ⊂ l2, a, s, a−, s− ⊂ l3.

There are eight invariant open triangles that are components of S2 −l1 ∪ l2 ∪ l3. Let 4 be one with vertices a, r, s. Then

(4∪ aro ∪ aso)/〈ϑ′〉, (4∪ aro ∪ rso)/〈ϑ′〉are both examples of compact annuli. The RP2-surface projectivelydiffeomorphic to these are said to be an elementary annulus.

Let 4′ be an adjacent triangle sharing aro with 4 in the boundary.Then

(4∪4′ ∪ aro ∪ aso ∪ as−o)/〈ϑ′〉, (4∪4′ ∪ aro ∪ rso ∪ rs−o)/〈ϑ′〉are examples of compact annuli again. The RP2-surfaces projectivelydiffeomorphic to these are said to be π-annuli. They are the union oftwo elementary annuli meeting at a boundary component. (See [16]and [17] for more details.)

Let S be the universal cover of an RP2-surface S with a developingmap dev : S → RP2 and a holonomy homomorphism h. For a circle S1,a closed geodesic c : S1 → S is a closed curve where dev c is a straightarc in RP2 for a lift c : R→ S of c. A closed geodesic is principal if fora lift c to the universal cover S, dev c is an embedding to a straightarc connecting an attracting and a repelling fixed point of h(γ) for adeck transformation γ of S satisfying c(t + 2π) = γ c(t). (Here γ

20 CHOI AND GOLDMAN

is said to be the corresponding deck transformation of c and c, and itexists uniquely.)

A properly convex domain in RP2 is a bounded convex domain of anaffine subspace in RP2. An RP2-surface is properly convex if it is a quo-tient of a properly convex domain in RP2 by a properly discontinuousand free action of a subgroup of PGL(3,R).

A disjoint collection of simple closed geodesics c1, . . . , cm decomposesan RP2-surface S into subsurfaces S1, .., Sn if each Si is the closure of acomponent of S −

⋃i=1,..,m ci where we do not allow a curve ci to have

two one-sided neighborhoods in only one Sj for some j.In [17], we proved:

Theorem 4.1. Let Σ be a closed orientable RP2-surface with principalgeodesic or empty boundary and χ(Σ) < 0. Then Σ has a collection ofdisjoint simple closed principal geodesics decomposing Σ into properlyconvex RP2-surfaces with principal geodesic boundary and of negativeEuler characteristic and/or π-annuli with principal geodesic boundary.

4.2. Complete hyperbolic surfaces and Fuchsian groups. Let Gbe a discrete subgroup of projective automorphisms of S+ in Aut(S). S+

has a complete Hilbert metric defined using cross-ratios and is isomet-ric with the complete hyperbolic plane, forming the so-called Beltrami-Klein model. (See [11].) We assume that G has no elliptic or parabolicelements. Here complete geodesics of the hyperbolic metrics are maxi-mal straight lines in S+ and vice versa.

We suppose that G is nonelementary: that is, it does not act on apair of points in Cl(S+). Then S+/G has a complete hyperbolic metricinduced from the Hilbert one. S+/G is geometrically finite in the sensethat it contains a compact surface bounded by closed geodesics that isthe deformation retract of it.

Let Λ denote the limit set of an orbit G(x) for a point x ∈ S+, whichis independent of the choice of x, as is well-known from the classicalFuchsian group theory. We know from the Fuchsian group theory thatΛ is a Cantor subset of bdSS+ and of Lebesgue measure zero and theset of fixed points of nonidentity elements of G is dense in Λ. Let usdefine Σ+ := S+/G. We define Σ′+ as Cl(S+) − Λ. (See Chapter 8 ofBeardon [4] for the classical Fuchsian group theory used here.) Let Jbe the set indexing the boundary components of Σ′. We denote theboundary components of Σ′+ by bi, i ∈ J .

Σ′+ = S+ ∪⋃i∈J

bi = Cl(S+)− Λ.


Then defining Σ′+ = Σ′+/G, we obtain a compact surface containingopen surface Σ+ compactifying it with boundary components diffeo-morphic to circles.

These surfaces are both examples of RP2-surfaces as they are quo-tients of open domains by projective automorphism groups.

Note that for each element g ∈ G, either g(bi)∩bi = ∅ or g(bi) = biholds with g in the cyclic group generated by the deck transformationcorresponding to bi and the boundary component of Σ′+ correspondingto it.

We obtain a convex domain Ω+ closed in S+ bounded by a unionof straight segments li for i ∈ J with same endpoints as bi. Thatis, li are geodesics in S+ connecting ideal endpoints of bi. Here, liis 〈gi〉-invariant for a unique primitive element of gi ∈ Γ where giacts along the orientation of li. Since Γ acts as a geometrically finiteFuchsian group on S+ without parabolics, the classical Fuchsian grouptheory tells us that Ω+/Γ is a compact hyperbolic surface with geodesicboundary that is the convex hull of S+/Γ and homeomorphic to Σ′+.Thus,

S+ − Ωo+ =

∐i∈J

Di

where Di is a convex open domain bdSDi = Cl(bi) ∪ li for i ∈ J thatcovers an open annulus in Σ′+.

Since li covers a simple closed geodesic in Σ′′+, it follows that

li = lj, i = j ∈ J or li ∩ lj = ∅, i, j ∈ J ,g(li) = li for g = gni , n ∈ Z or g(li) ∩ li = ∅, g ∈ Γ.(2)

4.3. The dynamic of geodesic flows on hyperbolic surfaces. Werecall that Γ is isomorphic to a free group of finite rank ≥ 2, andS+/Γ is a complete genus g hyperbolic surface with b ideal boundarycomponents. Assume that L(Γ) ⊂ SO(2,R)o. For convenience, denoteby Σ+ the interior of the surface Σ′+ from now on. That is, Σ+ = S+/Γ.

The Fuchsian Γ-action on the boundary bdSS+ of the standard diskS+ in S forms a discrete convergence group: For every sequence gj ofmutually distinct elements of Γ, there is a subsequence gjk and two(not necessarily distinct) points a, b in the circle bdSS+ such that

• the sequences gjk(x)→ a locally uniformly in bdSS+−b, thatis, uniformly in any compact subset of bdSS+ − b, and• g−1

jk(y) → b locally uniformly on bdSS+ − a respectively as

k →∞.

22 CHOI AND GOLDMAN

(See [1] for details.) We remark that

(3) a, b ∈ Λ.

For later purpose, we say that a is an attractor point and b is a repellorpoint of the sequence gjk.Lemma 4.2. Let gjk be a sequence satisfying the above convergencegroup properties. Suppose that the attractor point a is distinct from therepellor point b. Let ai and ri denote the attracting fixed point and therepelling fixed point of gi respectively. Then it follows that

(4) ajk → a and rjk → b.

Proof. One can use the fact that if a closed arc I is send to its interior Io

by a continuous map f , then the fixed points are in the image f(l). (Seealso ε-hyperbolicity in p.256 of [2]. Since a 6= b, these transformationsare uniformy ε-hyperbolic.)

Let US+ be the unit tangent vector bundle over S+, and let U(S+/Γ) =UΣ+ be the unit tangent vector bundle over S+/Γ = Σ+. By followingthe geodesics in Σ+, we obtain a geodesic flow

Φ : UΣ+ × R→ UΣ+.

A geodesic current is a Borel probability measure on U(S+/Γ) invariantunder the geodesic flow which is supported on a union of nonwanderinggeodesics. The subspace C(S+/Γ) composed of geodesic currents sup-ported on unions of closed geodesics is dense in the space of all geodesiccurrents with the weak-* topology.

Let UrecΣ+ denote the set of unit tangent vectors of nonwanderinggeodesics on Σ+. We recall

Lemma 4.3. Let Σ+ be as above. Then

• UrecΣ+ ⊂ UΣ+ is a connected compact geodesic flow invariantset and is a subset of the compact set UΣ′′+.• The inverse image UrecS+ of UrecΣ+ in UrecS+ is precisely the

set of geodesics with both endpoints in Λ.• The set of pairs of fixed points for all g ∈ G is dense in Λ×Λ.

Proof. The first item is in Lemma 1.2 in [35] and the second item isin the proof of the same lemma. The connectedness is also proved inLemma 1.3 in [35]. The third item follows from the fact that the set ofclosed geodesic orbits are dense in UrecΣ+ as the flow is Anosov.

Let ∆ denote the diagonal of Λ×Λ. Such pairs (λu, λs) ∈ Λ×Λ−∆correspond to nonwandering geodesics in the quotient hyperbolic sur-face. Thus, the set of closed geodesics is dense in the set of nonwan-dering geodesics by Lemma 4.3.


We have a compactification picture of the above phenomena: Fora nonelementary Fuchsian group Γ0 ⊂ PGL(2,R), the closure in theprojective space P

(Mat2(R)

)equals Γ0 ∪ (Λ× Λ), where Λ ⊂ bdSS+ is

the limit set of Γ0. Here we could write

P(Mat2(R)

)= PGL(2,R) ∪ (RP1 × RP1)

where RP1×RP1 corresponds to the set of projective equivalence classesof rank-one 2× 2 real matrices. (Projective equivalence classes of suchmatrices are completely determined by their kernels and images, whichare arbitrary lines through the origin in R2.) The kernel of a rankone 2 × 2 matrix is the undefined set for the corresponding singularprojective transformation, and the singular projective transformationis constant on the complement of this point in the one-dimensional realprojective space.

5. RP2-Schottky uniformizations

Goldman constructed an RP2-surface Σ with a free holonomy groupin [31]. The holonomy group is in the image PSO(2, 1) of SO(2, 1) inPGL(3,R). We will show that a domain in S regularly covers Σ.

As above, let Γ be a Margulis group without parabolics. (We willuse the notation of Section 4.2 letting G = Γ. ) We obtain an exactsequence

1→ Γ′ → Γ→ Z/2Z→ 1

where Γ′ is the subgroup of Γ of index ≤ 2 so that L(Γ′) ⊂ SO(2, 1)o.

5.1. The construction of the Γ-invariant domain. To begin, weassume L(Γ) ⊂ SO(2, 1)o initially and so Γ acts on S+. Then S+/Γ is acomplete genus g hyperbolic surface with b ideal boundary componentswhere b ≥ 1 and g is just some integer ≥ 0. In other words, S+/Γ canbe compactified by adding boundary components to a compact surfaceΣ′+ with b boundary components. The universal cover Σ′+ of Σ′+ isidentified with the union of a domain S+ and a collection of open arcsin bdSS+. (At the moment, we do not require that Γ acts properlydiscontinuously or freely on E. )

Let Λ denote the limit set of an orbit Γ(x) for x ∈ S+. We canidentify Σ′+ as Cl(S+) − Λ. Let J be the set indexing the boundary

components of Σ′. We denote the boundary components of Σ′+ bybi, i ∈ J .

Σ′+ = S+ ∪⋃i∈J

bi = Cl(S+)− Λ.

24 CHOI AND GOLDMAN

We will be using the same symbols for Γ its extension to S3 and itselement g and its extension to S3. We identify S with the boundaryof H , called a boundary sphere. Γ acts on the standard circles bdSS±and the interiors S± in S. Then each nonidentity element g of Γ has anattracting fixed point a in bdSS+, a repelling fixed point r in bdSS+,and a saddle-type fixed point s in S0. Of course, their antipodes inS are also an attracting fixed point a− in bdSS− and a repelling fixedpoint r− in bdSS− and a saddle-type fixed point s− in S0 respectively.

Since g acts on bdSS+ and fixes a and r, we obtain that g acts onthe two 1-dimensional subspaces tangent to bdSS+ at a and r. Hence,s, s− is their intersection. Therefore, we deduce that

(5) s, s− ⊂ ε(a) ∪ ε(r)as ε(a) and ε(r) are disjoint halves of these 1-dimensional subspaces.

We now denote by Σ′− the domain of antipodal points of Σ′+ in S−;

that is, Σ′− = A (Σ′). We note that Γ acts properly discontinuously

and freely on Σ′− by antipodality, and Σ′−/Γ is a compact RP2-surfacewith geodesic boundary diffeomorphic to Σ′+. Denote it by Σ′−. Also,let Ω− be the set of antipodal points to Ω+. Then Γ acts properly dis-continuously and freely on Ω+ and Ω−. Denote the respective compactquotient surfaces with geodesic boundary by Σ′′+ and Σ′′−.

For each point x of bdSS+, we recall that ε(x) is an open segment oflength π with end points x and −x tangent to bdSS+. The open arcsε(x) for x ∈ bdSS+ give leaves of the foliation F in S0. Let us give aninduced orientation on Σ′+ and hence an induced orientation on each lifor i ∈ J . Let pi and qi denote the forward and backward end points ofli. We draw a segment si = ε(pi). We also draw a segment ti = ε(qi).Then li, si, ti, and li,− bound an open disk Di invariant under 〈gi〉,which we call a strip. We denote by Ri the open strip union with liand li,−; that is,

Ri = Di ∪ li ∪ li,−, bdSDi = li ∪ si ∪ ti ∪ li,−.

(See Figures 1, 3, and 2 for the pictures of these arcs tangent to S+

and S−.)Using F , we obtain:

Proposition 5.1. The strips Ri and Rj are disjoint for i 6= j wherei, j ∈ J .

Proof. Since si, ti, sj, tj are all leaves of F corresponding to distinctpoints of bdSS+, the mutual disjointness of these arcs follows. (SeeFigure 3.) si does not intersect ti, sj, tj for i 6= j, i, j ∈ J , and ti doesnot intersect si, sj, tj for i 6= j. The elements li and li,− for i ∈ J


are mutually disjoint by (2). Thus the boundary bdSRi and bdSRj fori 6= j are disjoint. Hence, the conclusion follows.

s

t

i, −

l

l

i

i

i

Ri

Figure 3. The quotient spaces of the two “lune” regionsare π-annuli stereographically projected.

Proposition 5.2. Given Ri for i ∈ J and γ ∈ Γ, we either have

γ(Ri) = Ri or γ(Ri) ∩Ri = ∅.In the former case, γ = gni , n ∈ Z for the deck transformation gicorresponding to li. Furthermore, Ri/〈gi〉 is a compact annulus withgeodesic boundary that decomposes into two π-annuli or four elemen-tary annuli along some collection of simple closed geodesics.

Proof. Suppose that Ri ∩ γ(Ri) 6= ∅ holds. Then li ∩ γ(li) 6= ∅ holdssince the arcs ti are mutually disjoint. Hence, (2) implies the result.

There is a subspace l of dimension 1, a geodesic circle, containingli and li,− in S. l and ti bound a closed disk D1 and if we remove tiand the fixed points at the ends of li, this region covers a π-annuluswith principal boundary components. The similar statement is true forthe disk bounded by l and si. Therefore, Ai = Ri/〈gi〉 is a union oftwo compact π-annuli. In other words, l − Cl(li) − Cl(li−) maps to asimple closed principal geodesic decomposing Ai into two π-annuli withprincipal boundary. (See Figure 3.) Also, each π-annulus decomposesinto two elementary annuli. (See Section 4.1.)

We say that for i, j ∈ J , the annulusRi/〈gi〉 is equivalent toRj/〈gj〉if Rj = g(Ri) and ggig

−1 = g±j for g ∈ Γ. Thus, in fact, there are only

b equivalence classes of annuli of above form.

26 CHOI AND GOLDMAN

We define

Ai := Ri ∩ S0 =⋃x∈bi

ε(x) for i ∈ J .

We note that Ai ⊂ Ri for each i ∈ J . We finally define

Σ = Σ′+ ∪∐i∈J

Ri ∪ Σ′−

= Σ′+ ∪∐i∈J

Ai ∪ Σ′−

= Ω+ ∪∐i∈J

Ri ∪ Ω−(6)

= S−⋃x∈Λ

Cl(ε(x)).(7)

an open domain in S. Since the collection whose elements are of formRi mapped to itself by Γ, we showed that Γ acts on this open domain.

5.2. The RP2-surface. First, Σ does not contain any fixed point: Sup-pose that g ∈ Γ acts on li for some i ∈ J . Then g acts on Ri and thefour attracting and repelling fixed points are the vertices of Ri and twosaddle-type fixed points are in si or ti. Hence, they are outside Σ.

Suppose that g ∈ Γ − I does not act on any of the boundarycomponents. Then it has four attracting and repelling fixed pointsa, r ∈ bdSS+ and a−, r− ∈ bdSS−. By (5), saddle-type fixed points sand s− are either on a great segment ε(a) or the other one ε(r) tangentto bdSS+ at a and r respectively. Since a, r ∈ Λ holds, the fixed pointsare outside Σ by (7).

Let G ⊂ SO(2, 1)o be a subgroup whose elements preserve orientationon S+, and any open domain D an open domain on S upon which whereG acts freely and properly-discontinuously. Then D/G has an inducedorientation from S+.

Theorem 5.3. Let Γ be an orientation-preserving finitely generatedgroup in SO(2, 1)o without parabolics acting freely and properly discon-tinuously on S+ (or equivalent on S− ) isomorphic to a free group offinite rank ≥ 2. Let b be the number of ideal boundary components ofS+/Γ. Then

• there exists an open domain D in S where Γ acts properly dis-continuously and freely, and D/Γ is homeomorphic to a closedsurface of genus g, g ≥ 2.• As an RP2-surface D/Γ decomposes along simple closed princi-

pal geodesics into a union of b annuli A1, . . . , Ab and two convex


-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

Figure 4. A figure representing an actual free groupaction with some of the arcs si and ti drawn in point-plotson stereographically projected sphere as in Figure 2.

RP2-surfaces Σ′′+ and Σ′′−. Each annulus Ai, i = 1, . . . , b decom-poses into two π-annuli.• Γ also acts properly discontinuously and freely on D− := A(D),

and D−/Γ is a closed RP2-surface diffeomorphic to D/Γ.• D− also decomposes into Σ′′+ and Σ′′− and b annuli A′i.• The antipodal map A : S → S induces a projective diffeomor-

phism between D/Γ and D−/Γ.

Proof. We begin by letting D = Σ obtained as above. We showed thatΓ has no fixed point on D. Let ClD(Ai) denote the closure of Ai in Dfor each i ∈ J .

28 CHOI AND GOLDMAN

Now Γ acts properly discontinuously on Σ′+ and Σ′− and each g ∈ Γsends ClD(Ai) to ClD(Aj) for j 6= i or g is in the infinite cyclic sub-group of Γ acting properly discontinuously on ClD(Ai) if g(ClD(Ai)) =ClD(Ai). Hence, Γ acts properly discontinuously on

⋃i∈J ClD(Ai).

For any compact set K in Σ, we consider the compact sets

K1 := K ∩ Σ′+, K−1 := K ∩ Σ′−, and K0 := K ∩⋃i∈J

ClD(Ai)

since these are closed. We have K = K+ ∪K− ∪K0.Since each of ClD(Ai) has an open neighborhood Oj forming a mu-

tually disjoint collection of open sets covering K0, it follows that K0

meets only finitely many ClD(Ai).If g(K) ∩ K 6= ∅, we have g(Ki) ∩ Ki 6= ∅ for some i = −1, 0, 1.

Since Γ acts properly discontinuously on Σ′+ and Σ′− and⋃i∈J ClD(Ai)

respectively, there exist only finitely many elements g of Γ so thatg(Ki)∩Ki 6= ∅ for each i, i = −1, 0, 1. We showed that Γ acts properlydiscontinuously and freely on D.

Since Σ′+/Γ and Σ′−/Γ are compact surfaces, and⋃i∈J ClD(Ai)/Γ is

a union of finitely many compact annuli, it follows that D/Γ is a closedsurface.

For the second item, D/Γ decomposes along the images of the unionof li, li,−, i ∈ J to Σ′′+ and Σ′′− and annuli with boundary componentsthat are images of the union of li and li,−, i ∈ J . Proposition 5.2implies that these annuli are obtained from two π-annuli.

The third item follows since the antipodal map A conjugates Γ toitself and A (D) = D−. Let κ denote the induced diffeomorphismΣ→ D−/Γ where κ sends Σ′′− to Σ′′+ and vice versa. κ sends annuli inΣ to ones in D−/Γ but they do not share subdomains in the universalcover. (Actually Ri goes to its antipodal image which is distinct fromRi for each i ∈ J .)

The rest follows from this observation.

The proof of Theorem 1.1. Now, we consider the case when Γ is a sub-group of SO(2, 1) in general. Recall that Γ′ is the subgroup of Γ of index≤ 2 so that L(Γ′) ⊂ SO(2, 1)o. We construct D′ for Γ′ by Theorem 5.3.Let ψ be the element of Γ − Γ′. Then ψ is orientation-preserving butψ(S+) = S−. Since ψ preserves orientation, ψ acts on

∐i∈J Ai as con-

structed above for Γ′ as well. Thus, ψ acts on D′ and hence Γ acts onD′.

Recall from the introduction that L′(Γ) is a free group of rank ≥ 2acting freely on S+ as a subgroup of PGL(3,R) and S+/L′(Γ) is anonorientable complete hyperbolic surface. Since L′(Γ) acts freely on


S+, it follows that ψ goes to an element of PGL(3,R) that acts freelyon S+ and on

⋃i∈J bi. Hence, ψ as an element of Aut(S3) acts without

fixed point on ⋃i∈J

⋃x∈bi

ε(x) =∐i∈J

Ai

since ψ does not act on any fiber of the fibration. Hence ψ has no fixedpoint on

Σ′+ ∪∐i∈J

Ai ∪ Σ′−.

D′/Γ′ double-covers a closed surface D/Γ.Finally, it is straightforward to verify the same statements for A(D).

Replacing with this set is equivalent to replacing Cl(ε(x)) with Cl(ε(x)−)for every appropriate point x in (7).

The uniqueness follows from Theorem R of [31] since for a givenFuchian holonomy, a closed RP2-surface is obtained by constructioncalled grafting. Grafting is classified by a word of free semigroup ofrank two for each boundary component of Σ′+. We can easily showthat only two cases for each boundary component result in imbeddeddomains for the universal cover and developing maps in all other casesare immersions that are not imbeddings. (See also [18].)

6. The dynamic core of a Margulis spacetime

This section briefly summarizes the relevant results of Goldman,Labourie, and Margulis in [35] and [34].

6.1. Margulis invariants. Let Γ be a Margulis group without anyparabolic elements and with the property L(Γ) ⊂ SO(2, 1)o. Γ acts onS+ as a geometrically finite Fuchsian group.

An element g ∈ Γ−I is hyperbolic as it is not parabolic or elliptic.We choose so that v+(g), v0(g), v−(g) are in the standard orientationof R3; or v−(g) × v+(g) = v0(g). (See Definition 3.2.) We recall theMargulis invariant µ : Γ− I → R

µ(g) := B(gx− x, v0(g)) for g ∈ Γ− I, x ∈ E,

which is independent of the choice of x in E.If Γ acts freely and properly on E, then Margulis invariants of non-

identity elements are all positive or all negative by the opposite signlemma. (See [35].) By choosing the opposite orientation of E, we canchange the all signs of the invariants. We henceforth assume that theMargulis invariants of nonidentity elements are all positive.

30 CHOI AND GOLDMAN

In the second main theorem of [35], they extend the Margulis invari-ants to the geodesic current space on the hyperbolic space S+/Γ, thatis, diffused Margulis invariants for currents.

The Margulis invariants of elements of C(S+/Γ) are positive by thesecond main theorem in [35], and the subspace of currents supportedclosed geodesics with total measure 1 is precompact in C(S+/Γ). Itfollows that there exists a global C > 0 independent of g ∈ Γ so that

(8) 1/C ≤ µ(g)

lS+(g)≤ C for g ∈ Γ− I

where lS+(g) denotes the minimum of the hyperbolic distances betweenx and g(x) for x ∈ S+ where g is considered as an element of PGL(3,R)acting on S+ ⊂ RP2. (This equals the length of the closed geodesicrepresented by g. See also Corollary of [35].)

We remark that the positivity of the diffused Margulis-invariants isequivalent to the proper discontinuity of the action of Γ on E which isthe main result of Goldman, Labourie, and Margulis [35].

Goldman, Minsky, and Margulis [36] show that the positivity of theMargulus-invariants for closed curves is not sufficient to obtain theproper discontinuity of Γ on E by finding couter-examples when Σ+

is a one-holed torus but shows that the positivity of diffused Margulisinvariants laminations on surfaces is sufficient.

6.2. Neutralized sections. We note that Γ acts on US+ as a decktransformation group over UΣ+. An element γ ∈ Γ goes to the differ-ential map Dγ : US+ → US+. They in [34] constructed a flat affinebundle E over the unit tangent bundle UΣ+ of Σ+. They took the quo-tient E× US+ by the diagonal action given by γ(v, x) = (γ(v), Dγ(x))for a deck transformation γ ∈ Γ of the cover US+ of UΣ+.

The cover of E is denoted by E and is identical with E × US+. Wedenote by

πE : E = E× US+ → E

the projection.We define V as the quotient space of V × US+ under the diagonal

action defined by γ(v, x) = (L(γ)(v), Dγ(x)) for a deck transformationDγ of the cover US+ of UΣ+.

A neutral section of V is a SO(2, 1)-invariant section which is parallelalong geodesic flow of UΣ+. A neutral section ν : UΣ+ → V arisesfrom a graph of the SO(2, 1)-invariant map

ν : US+ → V


with the image in the space of unit spacelike vectors in V: ν is definedby sending a unit vector u in US+ to the normalization of the Lorentziancross-product ρ(u)×α(u) of the null vectors ρ(u) and α(u) respectivelyrepresenting the the starting point and the ending point in bdSS+ ofthe hyperbolic geodesic tangent to u in S+. (See Section 4.2 of [35] fordetails.)

Lemma 6.1. Suppose that u is a unit tangent vector to a geodesicwhere a deck transformation γ acts on along the orientation given byu. Then

• ρ(u) and α(u) respectively correspond in S+ a repelling fixedpoint and an attracting fixed point on bdSS+,• ν(u) is an eigenvector of L(γ) and corresponds to the saddle-

type fixed point of γ in S0.• γ acts on a unique spacelike line in E in the direction of ν(u) as

a translation by the vector µ(γ)ν(u) for the Margulis invariantµ.

Proof. Since the geodesic is invariant, the endpoints are attracting orrepelling fixed points. The Lorentzian cross-product of the attractingeigenvector and the repelling eigenvector is clearly another eigenvector.The final item is from an interpretation of the Margulis invariants.

Let ∇ be the flat connection on E as a bundle over US+ with fiberisomorphic to E induced from the product structure of E = E × US+.(See Sections 3.2 and 3.3 of [35].) We induce a flow

Φ : E× US+ × R→ E× US+

by letting the flow be constant in the second factor and be the geodesicflow on US+. This induces an Anosov flow

E× R→ E.

Using this Anosov geodesic flow of E in Lemma 8.4 of [35], Goldman,Labourie, and Margulis show that there exists a neutralized section,that is, a section N : UrecΣ+ → E satisfying

(9) ∇XN = fν

where X is the vector field of geodesic flow on UrecΣ+ and f is a realvalued function defined on UrecΣ+. Lemma 3 of [34] shows that we canchoose N so that f > 0 by changing N to N + gν for a suitable realvalued g : UrecΣ+ → R.

32 CHOI AND GOLDMAN

6.3. Lifting the neutralized section to the coverings. Let UrecS+

denote the inverse image of UrecΣ+ in US+. We take a compact funda-mental domain F in UrecS+ of UrecΣ+, and then N lifts on F to a mapN |F to E so that for each side pairing gS ∈ Γ of F for a side S of F ,

N gS(~u) = gS N (~u)

= (gS(πE N (~u)), gS(~u)) for every ~u ∈ F(10)

holds since N is obtained from an arbitrary lift satisfying this equationand deforming it (see the proof of Lemma 8.3 of [35]). Thus, we find

the section N : UrecS+ → E lifting N satisfying

(11) N γ = γ Nfor each deck transformation γ of US+ → US+.

Let u′ be a unit tangent vector to a closed geodesic in Σ+. Thegeodesic flow line α on UrecΣ+ tangent to u′ will return to u′ after aperiod. Suppose that a deck transformation γ in UrecS+ correspondsto a closed geodesic in Σ+, and acts on a geodesic lγ in S+ containing

a unit vector u. Let−→l γ denote the subspace of unit tangent vectors to

lγ in the γ-direction. Since N is a lift of N ,

πE N |−→l γ

maps to a line parallel to ν(u) up to an action of L(Γ) by (9). Sinceγ acts on the image of this line in E by (11), and L(γ) has ν(u) as aneigenvector by Lemma 6.1,

πE N |−→l γ

is a line parallel to ν(u).Equation (9) also lifts to

(12) ∇XN = f ν

where X is the unit-vector field of geodesic flow on US+, ∇ is the flatconnection on E× US+, and f is a positive-valued function. Thus, weconclude that the geodesic flow on UrecS+ is conjugated to one on theimage subset of UE by a map associated with πE N . (We note herethat the flow speed changes by the diffused Margulis invariants.)

6.4. Mapping the convex core. Theorem 1 of [34] shows that theimage of a geodesic γ of S+ mapping to a nonwandering geodesic in Σ+

under πE N is a spacelike geodesic in E mapping to a nonwanderinggeodesic in E/Γ.

Let UE denote the space of unit spacelike vectors at each point of Eand U(E/Γ) the space of unit spacelike vectors at each point of E/Γ.


Denote by UrecE in UE the inverse image of the subset of Urec(E/Γ)composed of unit tangent spacelike vectors tangent to nonwanderingspacelike geodesics.

Denote by −→yxo the open line segment connecting two points x, y ∈bdSS+ with the orientation towards x.

We give the topology of the space GrecS+ of oriented geodesics inS+ mapping to nonwandering oriented geodesics in Σ+ by the quotienttopology from the unit tangent bundle UrecS+. Similarly, we give thetopology of the space GrecE of spacelike oriented geodesics in E mappingto nonwandering geodesics in E/Γ by the quotient topology from thespace UrecE. (As usual, these are standard topological structures onsubspaces of complete geodesics in S+ and E.)

A bounded subset of GrecS+ is a set of geodesics passing a boundedset in US+, and a bounded subset of GrecE is a set of spacelike geodesicspassing a bounded set in UE under the Euclidean metric dE.

Proposition 6.2. The map N induces a continuous function N :GrecS+ → GrecE where

• For the attracting fixed point a and the repelling one r of g,

Axis(g) = N (−→rao).• the bounded set of elements of GrecS+ maps to a bounded set inGrecE.• Finally, the map is surjective.

Proof. For a flow segment in UrecS+, πE N sends it to a spacelike linesegment in E. This induces N . The first item is discussed in Section6.3.

The next statement follows by the fact that N is induced by thecontinuous map πE N which sends a compact set to a compact set.

Clearly our map is surjective from the set of geodesics in S+ map-ping to closed ones in S+/Γ to the set of geodesics in E mapping toclosed ones in E/Γ since any a closed geodesic correspond to a uniquenontrivial element of Γ in the one-to-one manner. The surjectivity ofN now follows by Theorem 1 of [34] by the density of closed geodesicsand the continuity of N .

7. Γ acts properly on E ∪ Σ.

Let Γ be a proper affine deformation of a convex cocompact Fuchsiangroup of rank ≥ 2. We assume initially that L(Γ) ⊂ SO(2, 1)o andS+/Γ is a complete genus g hyperbolic surface with b ideal boundarycomponents. As above, we assume that the Margulis invariants ofnonidentity elements are all positive.

34 CHOI AND GOLDMAN

-1.0-0.5

0.0

0.5

1.0

-1.0-0.5

0.00.5

1.0-1.0

-0.5

0.0

0.5

1.0

e3

e3−

e2e2−

η+

η−

e1

e1−

x

y

z

Figure 5. The action of a Lorentzian isometry g on the3-hemisphere H where the boundary sphere S is the unitsphere with center (0, 0, 0) here.

7.1. The action of projective-boost automorphisms. Recall thatS3 is the quotient R4 − O/ ∼ where ∼ is given by v ∼ w iff v = swfor s > 0. Recall that homogeneous coordinate system of S3 is given bysetting a point p of S3 to have coordinates [x : y : z : t] where (x, y, z, t)divided by its norm represent p as a unit vector in S3. We identified Ewith the open affine space given by t > 0. We use the coordinates sothat S corresponds to the hyperplane given by t = 0 and the origin Oof E is given by e0. The closure of E is the 3-dimensional hemisphereH with ∂H = S. For this system of coordinates, (x, y, z) ∈ E is givenhomogeneous coordinates [x : y : z : 1].


A projective automorphism g that is of form

(13)

λ 0 0 00 1 0 k0 0 1

λ0

0 0 0 1

λ > 1, k 6= 0

under a homogeneous coordinate system of S3 is a projective-boost. Inthis coordinate system, g acts on an open 3-dimensional hemisphere tobe identified with E given by t > 0, has an invariant axis Axis(g) givenby x = 0, z = 0, given by x = 0 and one U given by z = 0 containingit. On S, g contracts vectors along a direction by a constant 1/λ and gexpands the vectors along a direction by a constant λ on U assumingλ > 1. The subspace S given by x = 0 is the stable subspace, and thesubspace U given by z = 0 is the unstable subspace. (See Section 3.4.)

Let η+ be the segment e1e2e1−, “ the attracting arc”. Let η− denotethe segment e3e2−e3−, “ the repelling arc”. These have dS3-lengthsequal to π. The sphere S2

0 is called an unstable sphere.We will prove the following lemma for a properly convex K because

the proof is much simpler. But this is true for general compact subsetK.

Lemma 7.1. Let gλ,k denote the projective boost on H defined bythe above matrix in (13) for a homogeneous coordinate system withcoordinate functions t, x, y, z in the given order and let S2

0 denote thesubspace given by x = 0. We assume that k ≥ 0, λ > 0. Then asλ, k → +∞ where k/λ→ 0, the following statements hold:

(a) gλ,k|H − S20 converges in the compact open topology to a pro-

jective endomorphism Π0 given by sending [x : y : z : t] to e1 ore1− where the sign depends on the sign of x. (t ≥ 0 always.)

(b) gλ,k|(S20 ∩H ) − η− converges in the compact open topology to

a projective endomorphism Π1 given by sending [0 : y : z : t] toe2.

(c) For a properly convex compact set K in H −η−, we can choosea subsequence of gλ,k(K) that converges to either(i) e2 if K ⊂ Cl(U)− η−,

(ii) the point e1 or e1− if K ∩ Cl(U) = ∅,(iii) one of the segments

e1e2, e1−e2,

when K meets only one component of H − Cl(U), or(iv) η+ = e1e2e1−.

36 CHOI AND GOLDMAN

(d) Finally, the convergences here are uniform independent of Kprovided dS3(K, η−) ≥ δ in cases (ii), (iii), and (iv) or

dS3(K,Cl(U)) > δ

in case (i) for each fixed δ > 0.

Proof. The first two items follow by normalizing the above matrix bydiving it by the maximal norm of the entries.

For any properly convex compact subset K ′ in H − S20, gλ,k(K ′)

converges to one of e1 or e1− as λ, k → ∞ while k/λ → 0 bythe first item. Also, the case for K ′ ⊂ Cl(U), the proof is similar ask → +∞.

So, we now suppose that K ′ ∩ S20 6= ∅. Suppose first that K ′ meets

both components of H − S20. For a properly convex compact subset

K ′ in H − η−, let K ′1 be the nonempty properly convex compact setK ′ ∩ S2

0 − η−. We can take a closed 2-dimensional hemisphere H withboundary ∂H ⊂ S so that H is the topological boundary in H ofa convex open domain H ′ ⊂ H disjoint from η− and containing K ′

where we can freely and slightly enlarge subspaces and take joins. (Forexample, we can choose H to be a plane in E separating K ′ and η−containing e3 and e3− in the boundary.) Then the following statementshold:

• there exist a properly convex compact domain D1 in a compo-nent of H ′−S2

0 with x > 0 and another one D2 in a componentof H ′ − S2

0 with x < 0 so that K ′ is in the join J of D1 and D2

that can be chosen disjoint from η−.• For K1 := J ∩ S2

0 ⊃ K ′1, the join J1 of K1 and D1 and the joinJ2 of K1 and D2 satisfy J = J1 ∪ J2.

For example, we may obtain D1 and D2 by removing from K ′ an ε-dS3-neighborhood of S2

0 for small ε > 0 and making them slightly larger sothat the join of D1 and D2 contains K ′.

By the second item of Lemma 7.1, gλ,k(K1) converges to e2 asλ, k →∞. Since

gλ,k(D1) → e1 and gλ,k(D2) → e1−

as λ, k →∞ by above,

• gλ,k(J1) → s1 = e1e2 in S and• gλ,k(J2) → s2 = e1−e2 in S hold by Lemma 2.3 respectively.

Hence, we conclude gλ,k(J) → η+ as λ, k →∞.For each ε > 0,

gλ,k(K′) ⊂ Nε(η+) = Nε(s1 ∪ s2)


holds whenever λ and k are sufficiently large, and there are points arbi-trarily close to both points e1 and e1−. As K ′ is convex and connected,the geometric limit of gλ,k(K ′) exists as λ, k → ∞ and equals η+.(See Figure 5 also.)

So, we now suppose that K ′ ∩ S20 6= ∅ and that K ′ meets only one

component of H −S20. Then the arguments are simplier and gλ,k(K ′)

converges to e1e2 or e1−e2 as λ, k →∞.The final item (d) for cases (i) and (ii) can be obtained by easy

estimations. For (d)(iii) and (d)(iv), we apply the results for (i) or (ii)to D1 and D2 and K1. The uniform convergences for the joins J1 andJ2 follow by estimations also.

We remark that our proof of Proposition 7.2 is a generalization ofthe above one.

7.2. The proper discontinuity of the action of Γ. Recall fromTheorem 5.3 that we can construct

(14) Σ := S+ ∪∐i∈J

Ri ∪ S− ⊂ S,

that is a Γ-invariant open domain with an infinitely generated funda-mental group and covers a closed RP2-surface Σ, a union of Σ′+ and Σ′−and annuli Ai, i = 1, . . . , b.

Proposition 7.2. Let Γ be a Margulis group without parabolic ho-lonomy. Let Σ be as determined above by (14). Assume L(Γ) ⊂SO(2, 1)o. Then Γ acts freely and properly discontinuously on E ∪ Σas a group of projective automorphisms of S3.

Proof. Since Γ acts freely and properly discontinuously on R3 = E andso on the constructed Σ separately, Γ acts freely on the union.

Suppose that there exists a sequence gi of elements of Γ and acompact subset K of E ∪ Σ so that

(15) gi(K) ∩K 6= ∅ for all i.

We will show that actually the sequence gii∈Z+ has to be a finitesequence for the set of positive integers Z+.

We prove by contradiction: Suppose that gi is an infinite sequenceand gi 6= gj whenever i 6= j. Then (15) holds universally in this sectioneven if we take any subsequences.

Recall that Γ is a convergence group acting on bdS+. Let ai and ridenote the attracting fixed point and the repelling fixed point of gi re-spectively. We can choose a subsequence gjk satisfying the convergencegroup property (see Section 4.3). For the attracting fixed point ajk and

38 CHOI AND GOLDMAN

the repelling fixed point rjk in bdS+ of gjk , (4) shows ajk → a andrjk → b for a, b ∈ bdS+ provided a 6= b.

(I) First, we will consider the case when a 6= b:(I)(i) We collect some convergence properties of [35]: Now letting

gi denote the subsequence, we note that each gi has an attractingfixed point ai and a repelling fixed point ri in bdSS+. By our conditions,ai → a and ri → b hold by Lemma 4.2.

We define the Lorentzian cross-product

ν :=β × α|||β × α|||

of nonzero vectors α and β corresponding to a and b respectively. ALorentzian isometry element gi acts as a translation on a unique space-like line Axis(gi) in the direction νi of eigenvalue 1. Let αi and ρi denotethe null vectors in the directions of ai and ri respectively so that αi → αand ρi → ρ hold. We compute

νi :=ρi × αi|||ρi × αi|||

; i.e., νi = v0(gi).

Thus, [νi] → [ν] ∈ S holds. Since ai → a holds, the sequence

ai[νi]ai,− = Cl(ε(ai)) converges to a segment a[ν]a− = Cl(ε(a)). (Wecan assume without loss of generality that

|||ρi × αi||| = 1, |||β × α||| = 1.)

Since the geodesics with end points ai, ri pass the bounded part K ′′

of the unit tangent bundle of S+, it follows that

Cl(Axis(gi)) = Cl(N (−→riaio))→ Cl(N (−→bao))

by the continuity of N of Proposition 6.2. Hence, each Axis(gi) passesa point pi, and pi forms a convergent sequence in E. By choosinga subsequence, we assume without loss of generality that pi → p∞ forp∞ ∈ E.

Denote Ui := W wu(gi), which is the span of Axis(gi) and ri. Toconclude (I)(i), we obtained a sequence gi satisfying the properties :

αi → α, βi → β, νi = ρi × αi → ν,

ai → a, ri → b, [νi] → [ν], pi → p∞,

Cl(Axis(gi)) → [ν]p∞[−ν],

Ui := W wu(gi) → U∞.(16)

The last property follows easily. We will be fixing the property of giin this section here.


(I)(ii) The next important step of the proof is that the coordinatechanges required for gi so that it becomes of form in (13) are uniformlybounded:

We now introduce hi ∈ Aut(S3) coordinatizing S3 for each i. Wechoose hi so that

(17) hi(ai) = e1, hi([νi]) = e2, hi(bi) = e3, and hi(pi) = e4,

and Axis(gi) is sent to

e2e4e2−o.

By (16) and Lemma 2.1, hi can be chosen so that hi converges toa bi-Lipschitz map h ∈ Aut(S3), uniformly in Cs-sense for any integers ≥ 0.

We can consider v0(gi) = νi as a tangent vector at the origin O inE. The linearization L(hi) of hi restricted to E sends the unit-normvector v0(gi) to a vector parallel to (0, 1, 0). By post-composing witha projective map Φi fixing ei, i = 1, . . . , 4, we further modify hi sothat the differential of hi sends the tangent vector v0(gi) at pi to thetangent vector (0, 1, 0) at e4. Since v0(gi) is a convergent sequenceof vectors, we can still assume that hi is convergent in the above senseto h ∈ Aut(S3).

The sequence hi is uniformly bi-Lipchitz in dS3 ; There exists afixed positive contant C so that

C−1dS3(hi(x), hi(y)) ≤ dS3(x, y) ≤ CdS3(hi(x), hi(y))

C−1dS3(h−1i (x), h−1

i (y)) ≤ dS3(x, y) ≤ CdS3(h−1i (x), h−1

i (y))(18)

for all x, y ∈ Sn, i = 1, 2, . . . .The sequence of each quadruple of coordinate functions converges

to a quadruple of coordinate functions of a new adopted coordinatesystem h where

(19) h(a) = e1, h(b) = e3, h([ν]) = e2 and h(p) = e4

hold. Moreover, at the same time, we can assume that the linearizationof the affine map hi in E sends the vector νi to (0, 1, 0) in V and,similarly, the linearization of h sends the vector ν to (0, 1, 0) whichfollows as in the proof of Lemma 2.1 up to multiplying by a sequenceof bounded linear maps of E to the sequence.

We can conjugate by a sequence of hi of orientation-preserving bi-Lipschitz maps considered as an element of Aut(S3) so that higih

−1i is

40 CHOI AND GOLDMAN

of form

(20) gi :=

λ(gi) 0 0 0

0 1 0 k(gi)0 0 1

λ(gi)0

0 0 0 1

where λ(gi) > 1 and the orientation preserving hi is given by sendinge1 to ai, e2 to [νi], e3 to ri, and e4 to pi.

We note that

(21) gi = h−1i gi hi;

hence, gi acts like the standard form matrix in (20) up to coordinati-zation by hi.

Let us define the compact set Λ := H − (E∪ Σ), so-called stretched

limit set. Moreover, Λ =⋃z∈Λ Cl(ε(z)) for the limit set Λ of Γ. We

recall from Lemma 7.1:

• the unstable sphere S20 given by x = 0,

• the segment η+ := e1e2e1− and• the segment η− := e3e2−e3−.

Then

• h−1i (η−) = Cl(ε(ri)) is the segment ri[−νi]ri− ⊂ Λ and

• h−1i (η+) = Cl(ε(ai)) is the segment ai[νi]ai− ⊂ Λ.

(I)(iii) Now we estimate the entries of the above matrices.We claim that in this coordinate system hi

(22) µ(gi) = k(gi) :

we chose x in B(gi(x)−x, v0(gi)) to be on Axis(gi) and the linearizationL(hi) of hi restricted to E sends gi(x)− x and v0(gi) respectively to avector parallel to (0, 1, 0) and the vector (0, 1, 0). Since the Lorentziannorm ||v0(gi)|| = 1 holds, we obtain

gi(x)− x = µ(gi)v0(gi) and L(hi)(gi(x)− x) = µ(gi)(0, 1, 0).

k(gi) = L(hi)(gi(x)− x) · (0, 1, 0)

implies (22).Hence the positivity of the Margulis invariant is equivalent to k(gi) >

0 as the coordinate change is orientation-preserving where k(gi) de-pends on the coordinate system.

The hyperbolic metric on S+ gives us the closed geodesic ci in Σ+

represented by gi the hyperbolic length

lS+(gi) =

∣∣∣∣∣logλ(gi)

1λ(gi)

∣∣∣∣∣ = 2| log λ(gi)|,


where we used cross ratios.Recall that gi has a property (16):

Lemma 7.3. For gi obtained as above in (I)(i), and conjugating gi byhi as defined above to the form of (20), we have

(23) λ(gi)→ +∞, k(gi)→ +∞, andk(gi)

λ(gi)→ 0.

Proof. Since gi restricts to a sequence of mutually distinct hyperbolicisometries in S+ with sequences of fixed points ai → a and ri → band a 6= b, it follows that λ(gi) → ∞. (Suppose not. Then we seethat gi is a bounded subset of PSL(2,R) by conjugating ai and bi tothe standard position in bdS+ by a sequence of bounded matrices inPSL(2R). Since Γ is discrete and L′|Γ is injective, this implies gi isfinite. However, we assumed otherwise.)

By (8), we have µ(gi)→∞. Hence, the second limit follows by (22).By the main corollary of [35], µ(gi) ≤ C ′l(gi) and hence µ(gi) ≤

2C ′| log λ(gi)| hold; we obtain the final limit.

Note that without the right conjugating hi, we may not obtain theabove.

(I)(iv) We go back to our original compact set K from the beginningof the proof and cover it by compact convex sets that we can understandunder the action of gi: Recall from (I)(ii) that we may assume bychoosing subsequences of gi satisfying

• Ui → U∞, the limit unstable sphere a great 2-dimensionalsphere again,• h−1

i (η−) → η∞− = Cl(ε(b)) ⊂ Λ, and

• h−1i (η+) → η∞+ = Cl(ε(a)) ⊂ Λ.

Here

• η∞+ is a segment a[ν]a− ⊂ Λ and

• η∞− is a segment b[−ν]b− ⊂ Λ.

They are all tangent to bdSS+.Note that K is a subset of H −

⋃i∈Z+

h−1i (η−) since K ⊂ E ∪ Σ =

H − Λ. There are two components Ha and Ha− of H −U∞ containinga or a− respectively. The strategy is to cover K by balls bounded awayfrom U∞ and by balls that meets U∞ precisely at the centers. Thesetwo can be understood easily by the use of Lemma 7.1 and conjugationby hi that form a bounded uniformly convergent change of coordinates.We explain details:

(I)(iv-a) First, we try to obtain the covering. Since Ui convergesto U∞, we will cover K in the following manner:

42 CHOI AND GOLDMAN

• Let K∞ := K ∩ U∞. Assume that it is not empty. This is acompact subset of an open 3-dimensional hemisphere in U∞ −η∞− . We cover K∞ by the interiors Bo

i |i ∈ I1 of a finitely

many balls Bi ⊂ E ∪ Σ with center in U∞ for i ∈ I1 of smallradius ε0 > 0 in H − Λ. (We assume that balls Bi of radius 2ε0with the same centers are still in H − Λ.)• We split the balls and push the components B+

i , B−i of Bi−U∞

for each i ∈ I1 using a parameter of smooth maps preservingthe foliation in H by perpendicular segments to U∞ so that

– the images C+i ⊂ Ha, C

−i ⊂ Ha− are still compact and

convex,– for the join J(C+

i , C−i ) of the each pair of two halves C+

i , C−i ,

a so-called dumbbel, is still in Bi ⊂ E ∪ Σ, by displacingexclusively in the 2ε0-balls, and

– J(C+i , C

−i )o|i ∈ I1 covers K∞.

There is a positive number δ0 so that

dS3(C+i , U∞),dS3(C−i , U∞) > δ0 for all i ∈ I1 and(24)

K∞ ⊂⋃i∈I1

J(C+i , C

−i )o ⊂ E ∪ Σ.(25)

• We obtain a compact subset

K ′ := K −⋃

i=1,...,m0

J(C+i , C

−i )o ⊂ E ∪ Σ− U∞ = H − Λ− U∞.

Let εK′ := dS3(U∞, K′). We can cover the set by εK′/2-dS3-balls

Bi for i ∈ I2 where I2 is a finite set and

(26) dS3(Bi, U∞) ≥ εK′

2, i ∈ I2.

We divide I2 into I+2 and I−2 so that j ∈ I+

2 if and only if Bj ⊂ Ha.

We denote by C(0)j,i for j ∈ I1, the set Ui ∩ J(C+

j , C−j ).

(I)(iv-b) Now we study the balls that are bounded away from U∞.Since Ui → U∞,

• There exists N0 such that for i > N0, there exists δ > 0 suchthat

(27) dS3

Ui, ⋃j∈I+2

Bj ∪⋃j∈I−2

Bj ∪⋃j∈I1

C+j ∪

⋃j∈I1

C−j

> δ

by (24) and (26) since Ui → U∞.


• For i > N0, since we have that hi(Ui) = S20,

(28)

dS3

S20, hi(

⋃j∈I+2

Bj) ∪ hi(⋃j∈I−2

Bj) ∪ hi(⋃j∈I1

C+j ) ∪ hi(

⋃j∈I1

C−j )

> C−1δ.

• Now we act by gi. For any ε′′′ > 0, there exists N ′0 so that fori > N ′0,

(29)

dsS3

e1, e1−, gi hi

⋃j∈I+2

Bj ∪⋃j∈I−2

Bj ∪⋃j∈I1

C+j ∪

⋃j∈I1

C−j

< ε′′

holds by Lemma 7.1 (d).• Hence, for every ε′′ > 0, we have an integer N ′0 > N0 so that

for i > N ′0, the images of these sets under gi go into the ε′′-dS3-neighborhoods of a or a− by (18). That is,

(30) dsS3

a, a−, gi⋃j∈I+2

Bj ∪⋃j∈I−2

Bj ∪⋃j∈I1

C+j ∪

⋃j∈I1

C−j

< ε′′.

Thus, we conclude(31)

gi

⋃j∈I+2

Bj ∪⋃j∈I−2

Bj ∪⋃j∈I1

C+j ∪

⋃j∈I1

C−j

→ a, a−, a, or a−.

If K∞ = ∅, then the reader can proceed to (I)(iv-d) now using εK =dS3(U∞, K). Thus, assume K∞ 6= ∅ from now on.

(I)(iv-c) Now, we study the balls meeting U∞:We have

(32) dS3(⋃i∈I1

J(C+i , C

−i ), Λ) > δ′

for some δ′ > 0 since⋃i∈I1 Bi and Λ are both compact subsets of H

and are disjoint. Since Cl(ε(ri)) is a subset of the compact set Λ,

dS3(C(0)j,i ,Cl(ε(ri))) > δ′ for a fixed δ′ > 0.

Thus,

dS3(hi(C(0)j,i ), ηi−) > C−1δ′ > 0

holds for all j. Furthermore, the sequence of images under gi of

hi(C(0)j,i ) then converges to e2 by Lemma 7.1 uniformly with respect to

44 CHOI AND GOLDMAN

j. Hence, for every ε > 0,

dsS3(gi hi(C(0)j,i ), e2) < ε

holds if i > N ′′0 for some N ′′0 independent of j. By (18),

dsS3(gi(C(0)j,i ), [νi]) < Cε

holds. Therefore, the sequence gi(C(0)j,i )i∈Z+ converges to [ν] by

(21).Now, the sequence

• gi(J(Cj+, C

(0)j,i ))i∈Z+ → a[ν] by Lemma 2.3 and (30) and the

above convergence since

gi(J(Cj+, C

(0)j,i )) = J(gi(C

j+), gi(C

(0)j,i ))

for each i, j.

• Similarly, gi(J(Cj−, C

(0)j,i ))i∈Z+ → [ν]a−.

Since J(Cj+, C

j−) = J(Cj

+, C(0)j,i ) ∪ J(Cj

−, C(0)j,i ),

gi(J(Cj+, C

j−))i∈Z+ → η∞+ = a[ν] ∪ [ν]a− for each j ∈ I1

holds by Proposition 2.2.(I)(iv-d) We now come to the last step of the proof in case (I). If

gi(K)∩K 6= ∅ for all i, then there has to be some fixed pair of balls Band B′ in the above finite collection I1∪I2 of balls so that gi(B)∩B′ 6= ∅holds for infinitely many i. However,

gi(B) → η∞+ , a, or a− ⊂ S with η∞+ ∩ Σ = ∅.

Thus, for each ε > 0 and every ball B in the collection I1 ∪ I2, thereexists an integer I0 such that

Nε(η∞+ ) ⊃ gi(B) if i > I0

by definition of the geometric convergence. However, Nε(η∞+ ) ∩B′ = ∅

holds for any ball B′ in I1 ∪ I2 for sufficiently small ε, and we obtain acontradiction.

(II) Now we consider the a = r case. Choose an element γ0 ∈ Γ sothat γ0(a) 6= a as Γ is not an elementary Fuchsian group acting on S+.

Consider the sequence γ0gi. For each ε > 0 and sufficiently large iand a precompact neighborhood U in bdSS+ − a.

γ0gi(U) ⊂ Nε(γ0(a))


holds. Now consider the sequence g−1i γ−1

0 . Let V be a precompactneighborhood in bdSS+−γ0(a). Since γ−1

0 (V ) is a precompact inter-val in bdSS+ − a and for arbitrary ε > 0,

g−1i (γ−1

0 (V )) ⊂ Nε(a) for i > I0

holds for some large I0. Thus, γ0(a) is the attractor point of γ0giand a is the repellor point.

Since γ0gi(K ∪ γ0(K)) ∩ (K ∪ γ0(K)) 6= ∅ for infinitely many i, weare reduced to the case where a 6= r by replacing gi with γ0gi andK with another compact set K ∪ γ0(K) ⊂ E ∪ Σ. Again, we obtaincontradiction.

Remark 7.4. Let Γ be given an equivalence relation ∼ of conjugacy.We can replace the condition of the proper-discontinuity of the actionof Γ in E by the condition (*) given by

µ(g) > 0 for g ∈ Γ− Iµ : Γ− I/ ∼→ [0,∞) is proper, and

N is continuous.

which seemingly is slightly weaker to the condition of (8). Then we canshow that this condition implies that Γ acts properly discontinuouslyon E ∪ Σ.

Let Γ be a free group of rank ≥ 2 acting freely on E without parabol-ics. We can show the equivalence of the following conditions:

(a) Γ acts properly and freely on E.(b) The Margulis invariants of Γ satisfy equation (8) and N is

continuous.(c) Our condition(*).(d) Γ acts properly and freely on E ∪ Σ.

(b) ⇒ (c) and (d) ⇒ (a) are trivial. (a) ⇒ (b) is in [35], and (c) ⇒(d) is in the proof of Proposition 7.2.

Remark 7.5. In the proof of Proposition 7.2, once we obtain some cov-ering of K by compact convex balls, instead of the above precise ones,then we can just use Lemma 7.1 and using how Ui intersect with eachof the convex balls and extracting subsequences, we can still obtainthe proof. This is the shorter proof that the authors used in the earlierpreprint.

46 CHOI AND GOLDMAN

8. The proof of tameness.

Using the above notations, we note that Σ/Γ is a closed surface ofgenus g and forms the boundary of the 3-manifold M := (E ∪ Σ)/Γby Proposition 7.2 provided L(Γ) ⊂ SO(2, 1)o. Therefore, M is a 3-manifold in general since Γ has an index ≤ 2 subgroup Γ′ with theproperty L(Γ′) ⊂ SO(2, 1)o and the fact that Γ acts freely on E ∪ Σ.

We now show that M is compact. The key idea is to isotopy somespheres into E and they have to bound compact 3-balls. We first assumethat L(Γ) ⊂ SO(2, 1)o so that Γ acts on S+ honestly.

Proposition 8.1. Each simple closed curve γ in Σ bounds a simpledisk in E∪ Σ. Let c be a simple closed curve in Σ that is homotopicallytrivial in M . Then c bounds an imbedded disk in M .

Proof. This is just Dehn’s lemma.

We can find a collection of disjoint simple curves γi, i ∈ J ′, on Σ foran index set J ′ so that the following hold:

•⋃i∈J ′ γi is invariant under Γ.

•⋃i∈J ′ γi cuts Σ into a union of open pair-of-pants Pk, k ∈ K, for

an index set K. The closure of each Pk is a closed pair-of-pants.• Pkk∈K is a Γ-invariant set.• Under the covering map π : Σ→ Σ/Γ, each γi for i ∈ J ′ maps

to a simple closed curve in a one-to-one manner and each Pkfor k ∈ K maps to an open pair-of-pants as a homeomorphism.

This is achieved by finding arcs in S+/Γ cutting it into disks. Recall-ing that S+/Γ is an open surface compactified to the compact surfaceΣ′+ with boundary, we obtain a system of geodesic segments αi, i ∈ Ifor a finite set I, Σ′+ cutting it into a union of disks each of which isbounded by six arcs, alternating triple of which are arcs coming from∂Σ′+ and the other alternating triples are geodesic segments ending∂Σ′+. We can assume that each αi is imbedded in S+/Γ cutting it intofinitely many hexagons so that each ideal boundary component of S+/Γmeets αi and αj for some pair i, j, i 6= j.

This gives a system of geodesic arcs αj, j ∈ J ′, in S+ for someinfinite index set J ′ which decomposes S+ into hexagons and forms aΓ-invariant set. We define αj− := A(αj) in S− for j ∈ J ′. We connecteach endpoint δ1(αj) and δ2(αj) of αj with its antipodal endpoint ofαj− by ε(x).

γj := αj ∪ αj− ∪ ε(δ1(αj)) ∪ ε(δ2(αj)), j ∈ J ′.


(These form crooked circles.) We do this for each arc and obtain theabove system of simple closed curves. By construction, γii∈J ′ mapsto a system of disjointly imbedded curves γ1, .., γ3g−3, and each of them

has a simple closed lift in Σ and hence has a trivial holonomy. See Fig-ure 6. Also, the collection γ1, . . . , γ3g−3 decomposes the closed surface

Σ/Γ of genus g for some g ≥ 2 into 2g− 2 pairs of pants P ′1, . . . , P′2g−2.

(Here g depends on the genus g of S+/Γ and the number of idealboundary components.)

By trivial holonomy and Dehn’s lemma, each γi bounds a disk Di

in M . By the 3-manifold topology of disk exchanges, we can chooseD1, . . . , D3g−3 to be mutually disjoint. (See [38].)

By combinatorics of the situation, the Cauchy completion of eachcomponent of M − D1 − · · · − D3g−3 is homeomorphic to a compact3-ball in as it is bounded by a 2-sphere that is a union of a pair-of-pants P ′j and three of D1, . . . , D3g−3: We can lift the sphere into E∪ Σand being a sphere which can be pushed inside E, it bounds a compact3-ball disjoint from other disks.

The image of the 3-ball is a compact 3-ball in M as well. Since Mis a union of the closure of the components that are 3-balls identifiedwith one another in disjoint disks, M is a compact 3-manifold. Thisimplies also that M is homeomorphic to a solid handlebody of genusg; that is, our manifold (E ∪ Σ)/Γ is a compact 3-manifold and so itsinterior E/Γ is tame. This completes the proof of Theorem 1.2 providedL(Γ) ⊂ SO(2, 1)o. (Also, it follows that the rank of Γ equals g.)

In the general case when L(Γ) is not a subgroup of SO(2, 1)o, theabove work applies to (E∪ Σ)/Γ′ double-covering (E∪ Σ)/Γ. By Theo-rem 5.2 of Chapter 5 of [38], M is homeomorphic to a handlebody sinceΓ is a free group and (E ∪ Σ)/Γ is an aspherical compact 3-manifold.

A crooked plane is a closed disk embedded in E constructed as follows:Take two null vectors v1 and v2 and respective parallel lines L1 and L2

meeting at a point x. Then they are on a timelike plane P . Take alltimelike lines on P from x and take their union. It is a union of twocones C1 and C2 with vertex at x. We obtain two null half-planes W (v1)and W (v2). A crooked plane is the union C1 ∪ C2 ∪ W (v1) ∪ W (v2).(See [21] for details.)

A disk in E is an almost crooked plane if it agrees with a crookedplane in the complement of its compact subset. Also, its immersed orimbedded image is said to be crooked plane as well.

Proposition 8.2. Let Σ and Γ be as above. The Margulis spacetimeE/Γ with a free fundamental group has a system of disks with boundary

48 CHOI AND GOLDMAN

Figure 6. The arcs in S+ and an example of γi in thebold arcs.

in E ∪ Σ/Γ so that the closures of components of the complement arecompact 3-balls and the disks are almost crooked planes.

Proof. First, assume L(Γ) ⊂ SO(2, 1)o. Each of our disks has theboundary of a crooked plane. In the Dehn’s lemma, one can arbitrarilyassign the tubular neighborhood of the boundary of each disk as longas it is transversal to Σ/Γ. Thus, following the above discussions, weobtain the proof in this case.

Suppose that L(Γ) is not in SO(2, 1)o. Then we take a quotientΣ′ := S+/L′(Γ) given by the projective action. The nonorientable Σ′

admits a decomposition into hexagons as above. We obtain the induceddecomposition on S+∪S− and we extend the above construction to thissituation and obtain the crooked circles and crooked planes.

As above E∪Σ/Γ′ double-covers E∪Σ/Γ with the deck transformationgroup generated by a projective automorphism φ of order two. Bychoosing a collection for the quotient S+/L′(Γ) as in the beginning ofthe proof of Proposition 8.1, we may assume without loss of generalitythat φ sends each of the above crooked circles to itself or to a disjointone by our construction. We can modify each almost crooked planeD not changing a neighborhood of ∂D in D so that D ∩ φ(D) = ∅using the proof of Theorem 3 of [37]. Hence, we obtain a collection ofφ-equivariant almost crooked planes that cuts M into 3-balls.

Corollary 8.3. A Margulis spacetime without parabolics has a finitesided fundamental polyhedron in E.


Proof. We take a union of finitely many of the 3-balls obtained bycutting along the disks in E.

9. Asymptotics of Margulis spacetimes

Let N be a submanifold in a projective manifold M and is closed. Ifeach boundary point of N has a convex open neighorborhood B whereB − B ∩ N is a convex open domain, then N is said to have concaveboundary. Let Γ be a Margulis group without parabolics satisfyingL(Γ) ⊂ SO(2, 1)o. We show the existence of two convex hypersurfacesin E asymptotic to bdSS+ and bdSS− respectively. We will use thisto show that a concave boundary submanifold contains all the closedspacelike geodesics in E/Γ in the next section.

9.1. Supporting hyperplanes. LetH be the set of pairs (x, h) wherex is a point of the limit set Λ in bdSS+ and a 2-dimensional hemi-sphere h in H containing x in its boundary ∂h. We give the Hausdorffconvergence topology to this space. γ in Γ acts by sending (x, h) to(γ(x), γ(h)).

Proposition 9.1. For each point x of the limit set Λ of Γ in S inbdSS+, there exists a unique 2-dimensional hemisphere hx dependingcontinuously on x so that hx contains the fixed line Axis(g) ⊂ R3 of anelement of g ∈ Γ if x is an attracting fixed point of g. Moreover, thereexists a map τ : Λ→ H sending x to (x, hx) that is continuous and themap is Γ-equivariant ; that is,

τ(γ(x)) = (γ(x), γ(hx)) for τ(x) = (x, hx), γ ∈ Γ.

Proof. If x is an attracting fixed point of γ ∈ Γ, we let hx be the 2-hemisphere containing the invariant line Axis(γ) ⊂ E of γ and x as thestabilizer subgroup of Γ is infinite cyclic and is uniquely determined byx by the elementary Fuchsian group theory. Then hx is γ-equivariantand ∂hx contains x, x−, s, s− in S for the saddle-type fixed points s ands− and the antipode x− of x. Thus, τ is defined for the set of attractingfixed points of Γ.

The inverse image in US+ of Urec(S+/Γ) is precisely the set of geodesicswith endpoints in Λ. (See Lemma 4.3 and the proof of Lemma 1.3 of[35].) Recall that Λ∗ = Λ×Λ−∆ consists of elements (x, y) where x isthe ending point and y is the starting point of a nonwandering geodesicl.

For convenience, we use the notation from Proposition 6.2

L(x, y) := N (−→yxo)

50 CHOI AND GOLDMAN

with the orientation forgotten. Using Proposition 6.2, we define a func-tion τ(x, y) : Λ∗ → H by

(x, y) 7→ (x, h(x, y))

for the hemishphere h(x, y) = H ∩S2 for the span S2 of x and L(x, y).Note here that L(xg, yg) = Axis(g) if xg and yg are fixed points of g inbdSS+ ordered appropriately, and h(x, y) is contained in the unstablesubspace of g.

Notice that τ(a, y) is independent of y, y ∈ Λ−a if a is an attract-ing fixed point of an element of Γ with the repelling fixed point r in Λand equals τ(a, r) by the main result of Charette, Goldman, and Jones[14]: L(a, y) has to lie in the unstable plane h(a, r)o the line L(a, r)going to a closed geodesic in L(a, y) has to go to a line spiralling to theclosed geodesic in E/Γ. This fact is subsumed under Lemma 9.3.

We state a fact we need.

Lemma 9.2. Let (xi, yi) ∈ Λ∗ be a sequence. Suppose that xi →x, yi → y. Then

L(xi, yi) → L(x, y), τ(xi, yi) → τ(x, y) as i→∞.Proof. This follows by the fact that N is continuous by Proposition6.2.

Lemma 9.3. Let (x, y) ∈ Λ∗. Then τ(x, y) is independent of y ∈Λ− x, and h(x, y) 6⊂ S for every (x, y) ∈ Λ∗.

Proof. Suppose that x is an endpoint of two nonwandering geodesics lwith the starting point y and another one l′ with the starting point y′.

If (x, y) ∈ Λ∗, then L(x, y) is a line in E by Proposition 6.2; h(x, y)is not in S.

Recall that Σ′′+ = Ω+/Γ is a compact hyperbolic surface with geodesicboundary and the nonwandering geodesics are always in it. Choose asequence pi of points xyo so that pi → x. There exists a sequenceγ−1

i of elements of Γ so that γ−1i (pi) ∈ F for a fixed compact funda-

mental domain F of Ω+ under Γ.By choosing a subsequence, we may assume that pi → p∞ ∈ F

and γ−1i (Cl(l)) converges to a segment s∞ passing p∞ and ending

at two points of Λ. Let l∞ denote the interior of s∞ in Ω+. Let x∞and y∞ denote the endpoints so that we may assume without loss ofgenerality that γ−1

i (x) → x∞ and γ−1i (y) → y∞.

Since Γ is not elementary, we may assume that x, y∩x∞, y∞ = ∅by multiplying by γ−1

i ∈ Γ if necessary.Since γ−1

i (x) → x∞ and γ−1i (y) → y∞,

Cl(L(γ−1i (x), γ−1

i (y))) = γ−1i (Cl(L(x, y))) → Cl(L(x∞, y∞))


holds by continuity of N by Proposition 6.2. Hence,

(33) γ−1i (h(x, y)) = h(γ−1

i (x), γ−1i (x)) → h(x∞, y∞)

holds.By using perpendicular geodesics in the hyperbolic space S+ to the

geodesic l, we can show that y∞ is the attractor point of the sequenceγ−1

i since γ−1i sends any precompact open set in bdSS+ − x to an

arbitarily small neighborhood of y∞ for sufficiently large i. Also, x isthe repellor point of the sequence γ−1

i since γi sends any precompactopen set in bdSS−−y∞ to an arbitrarily small neighborhood of x forsufficiently large i using geodesics perpendicular to γ−1

i (l). Therefore,for an attracting fixed point ai and repelling fixed point ri of γ−1

i foreach i, it follows that ai → y∞ and ri → x by Lemma 4.2 sincex 6= y∞. riai → xy∞ implies

Cl(Axis(γ−1i )) = Cl(L(ri, ai)) → Cl(L(x, y∞)),(34)

h(ri, ai) → h(x, y∞).(35)

From this, y′ 6= x and y′ ∈ bdSS+ show

γ−1i (y′)→ y∞,(36)

γ−1i (h(x, y′)) = h(γ−1

i (x), γ−1i (y′)) → h(x∞, y∞).(37)

Since ∂h(x, y) is a great circle tangent to bdSS+ at x and so is∂h(x, y′), we obtain ∂h(x, y) = ∂h(x, y′). And γ−1

i (∂h(x, y′)) is tan-gent to bdSS+ at γ−1

i (x). Thus, γ−1i (∂h(x, y′)) converges to a great

circle S1x∞ tangent to bdSS+ at x∞ in S.

Choose a point q in the interior of the 2-hemisphere correspondingto h(x, y′) bounded away from ∂h(x, y) = ∂h(x, y′):

dS3(q, ∂h(x, y)) ≥ C0, C0 > 0.

Suppose that q ∈H −h(x, y∞) for contradiction. Then q is uniformlybounded away from h(ri, ai) for i > I0 for a fixed integer I0 by (34).That is, there exists a constant C1, C1 > 0, such that

dS3(q, h(ri, ai)) > C1.

We claim that γ−1i (q) → y∞ or γ−1

i (q) → y∞,−:We write γ−1

i in the matrix of the form of (13) with respect to co-ordinate change automorphisms hi ∈ Aut(S3) so that hiγ

−1i h−1

i is ofform

(38) γ−1i :=

λ(γ−1

i ) 0 0 00 1 0 k(γ−1

i )0 0 1

λ(γ−1i )

0

0 0 0 1

52 CHOI AND GOLDMAN

where λ(γ−1i ) > 1, k(γ−1

i ) > 0 and

hi(ai) = e1, hi(si) = e2hi(ri) = e3, hi(pi) = e4,

where we choose pi ∈ Axis(γ−1i ) so that pi is uniformly bounded away

from S by (34), and si is the saddle-type fixed point or the point corre-sponding to an end point of Axis(γ−1

i ) in S. As above, hi is uniformly bi-Lipschitz with respect to dS3 and can be assume to be convergent. Also,h(ri, ai) is the image of the 2-hemisphere in H given by S2

0 of Lemma7.1 under h−1

i . (We are exactly in the situation of the part (I) of theproof of Proposition 7.2 in Section 7.) As before, k(γ−1

i ) ≤ C log λ(γ−1i )

holds for the terms in the matrices of γ−1i by Lemma 7.3. By the first

item of Lemma 7.1, we obtain

γ−1i (q) → y∞ or γ−1

i (q) → y∞−

since hi are uniformly bi-Lipschitz and

dS3(q, h(ri, ai)) > C1 for i > I0 and a constant C1 > 0.

The proof is entirely same as in the part (I)(iv-b) of the proof of Propo-sition 7.2.

Since γ−1i (q) → y∞ or y∞− ∈ S and γ−1

i (∂h(x, y′)) convergesto a great circle S1

x∞ tangent to bdSS+ at x∞ in S, it follows that

γ−1i (h(x, y′)) geometrically converges to a 2-hemisphere in S by Lemma

2.3 up to a choice of subsequence. We obtained a contradiction to (36).Therefore, q ∈ h(x, y∞) and h(x, y′) = h(x, y∞) by geometry.Similarly, we have h(x, y) = h(x, y∞) by (33), and we conclude that

h(x, y) = h(x, y′).

Now we complete the proof of Proposition 9.1: We have defined thefunction τ : Λ→ H and the 2-hemisphere h(x) for any x ∈ Λ by

τ(x) := τ(x, y), h(x) := h(x, y) for any y ∈ Λ− x.Now, we show that τ is continuous.

Now suppose that xi is a sequence of arbitrary points in Λ so thatxi → x. We choose yi ∈ Λ for each i so that yi → y ∈ Λ fory 6= x. Then h(xi, yi) → h(x, y) holds by Lemma 9.2 and therefore,τ(xi) → τ(x) holds by the independence of Lemma 9.3. Therefore,τ is a continuous function.

9.2. Domains. For each i ∈ J , let4i be an open gi-invariant triangledisjoint from Ω+ with an edge li where gi is a corresponding element ofΓ acting on li. Hence its attracting and repelling fixed points ai, ri, atthe ends and a saddle-type fixed point si are vertices of 4i. We chose


the triangle 4i ⊂ Ri for the rectangular domain Ri defined in Section5.

In S, at each point x of Λ, there is a closed 2-hemisphere Hx ⊂ Ssupporting the convex hull of Λ. We form an intersection D(Λ) =⋂x∈ΛH

ox ⊂ S. Then D(Λ) is an open properly convex domain in S:

(39) D(Λ) =⋃i∈J

4i ∪ S+ ⊂ Σ.

Hence, D(Λ) ⊂ Σ holds. Moreover, D(Λ)/Γ is an open surfaceobtained from Σ+ by adding open elementary annuli 4i/〈gi〉 in corre-sponding annuli Aj. Thus, the closure of D(Λ)/Γ in Σ is a compactsurface in Σ with boundary components cj ⊂ Aj for each j = 1, . . . , b.

Defining D(A (Λ)) = A (D(Λ)), we obtain closed curves c′j ⊂ Aj foreach j = 1, . . . , b that are boundary components of D(A (Λ))/Γ in Σ.Note that (Σ − D(Λ) − D(A (Λ)))/Γ is a union of compact annuli ineach of Aj, j = 1, . . . , b. We denote each component annulus by Bj sothat Bj ⊂ Aj where ∂Bj = cj∪c′j. (Each Bj is a sum of two elementaryannuli and is bounded by nonprincipal boundary components. )

Lemma 9.4. Let NΓ be the set of points where the nonwandering space-like geodesics passes in E/Γ. Then for any finite index subgroup Γ′′ ofΓ, the natural map

NΓ′′ ⊂ E/Γ′′ → NΓ ⊂ E/Γ

is a finite covering map.

Proof. If the ω-limits of a spacelike geodesic l in E/Γ′′ is compact, thenso is the one for its image geodesic l in E/Γ. Conversely, the inverseimage of the compact ω-limit of a spacelike geodesic l in E/Γ has acompact inverse image and the lifting spacelike geodesic l must havethe ω-limit in it. The same argument works for α-limits. Thus, for acovering p1 : E/Γ′′ → E/Γ, we obtain p−1

1 (NΓ) = NΓ′′ , which impliesthe conclusion.

The boundary of a convex open domain in E is asymptotic to a setK in S+ if every unbounded sequence of points in it has all limit pointsin K.

Corollary 9.5. Let Γ be a Margulis spacetime without parabolics. InE, there exists a Γ-invariant nonempty convex open domain D whoseboundary in E is asymptopic to bdSD(Λ), homeomorphic to a circle.There exists another Γ-invariant convex open domain D′ whose bound-ary in E is asymptotic to A (bdSD(Λ)) so that the closures of D and

54 CHOI AND GOLDMAN

D′ are disjoint. Moreover, every nonwandering spacelike geodesic iscontained in a manifold

(E−D −D′)/Γ

with concave boundary.

Proof. First, we work with the index ≤ 2 sugroup Γ′ of Γ with L(Γ′) ⊂SO(2, 1)o. Let Λ denote the limit set of Γ′. For each x ∈ Λ, take theopen half-space Hx ⊂ E bounded by h(x, y) for some y ∈ Λ − xand containing Λ in its closure in the 3-hemisphere H . Then we takeD :=

⋂x∈ΛHx. The domain is a properly convex open domain since it

has at least three supporting hyperplanes and S in a general position.We show that D is not empty: That is, we claim that there exists a

compact convex set K ⊂ E so that Hx∩K 6= ∅ for all x ∈ Λ and hencethere exists

z ∈ K ∩⋂x∈Λ

Cl(Hx)

by compactness. Then the join J(z, bdSS+) is a subset of Cl(Hx) forevery x ∈ Λ. Hence the topological interior of the join J(z, bdSS+) isa nonempty subset of D.

Suppose that such a compact set K does not exist. Then thereexists a sequence of 2-hemispheres h(xi, yi), xi, yi ∈ Λ so that it boundsan open half-space Hxi in E and Hxi ∩ K = ∅ for each compactset K ⊂ E if i > IK for an integer IK depending on K. Since Λ iscompact, this is equivalent to the statement that h(xi, yi) convergesto a 2-hemisphere in S up to a choice of a subsequence. Let x∞ be alimit point of the sequence xi. Then τ(x∞) is a pair (x∞, h∞) for a2-hemisphere h∞ in S by the continuity of τ contradicting Lemma 9.3.

Let bdE(⋂x∈ΛHx) be the topological boundary in E. The closure of

D =⋂x∈ΛHx is a convex compact subset of S3, and

bdCl(D) = Cl(D(Λ)) ∪ bdE(⋂x∈Λ

Hx).

Thus, every unbounded sequence of points in bdE(⋂x∈ΛHx) will have

limit points in D(Λ). Since the limit points cannot be in D(Λ)o, theyare in bdSD(Λ).

For each Hx, we can take (E − Hx)o and take the intersection over

all x ∈ Λ. This gives us

D′ :=⋂x∈Λ

(E−Hx)o,


which is again a properly convex open domain and is nonempty. Also,

bdD′ = Cl(A(D(Λ))) ∪ bdE

⋂x∈Λ

(E−Hx)o.

These follow as before for D.First, notice D ∩ D′ = ∅. If Cl(D) ∩ Cl(D′) 6= ∅ holds, then it is a

compact convex subset of E as the boundary lies in S and hypersurfacesin E and the fact that

Cl(D(Λ)) ∩ Cl(A (D(Λ))) = ∅.Since Cl(D)∩Cl(D′) is also Γ′-invariant, the Brouwer fixed-point theo-rem implies a contradiction that the intersection contains a fixed pointfor each element g ∈ Γ′. Hence, Cl(D) ∩ Cl(D′) = ∅.

Every nonwandering spacelike geodesic equals L(x, y) for (x, y) ∈ Λ∗

by Theorem 1 of [34]. Since L(x, y) is in h(x, y) for some (x, y) ∈ Λ∗,and h(x) ∩ (D ∪D′) = ∅ holds, the result follows provided Γ = Γ′.

Suppose now that Γ − Γ′ 6= ∅. Let φ ∈ Γ − Γ′. Then φ induces aninner automorphism g 7→ gφ = φgφ−1 : Γ′ → Γ′. Thus, if (x, h(x)) isfixed by g, then (φ(x), φh(x)) is fixed by gφ with φ(x) in bdSS− andis uniquely determined by gφ. We obtain φ(h(x)) = h(A(φ(x))) sinceφ(h(x)) contains Axis(gφ) = φ(Axis(g)). Thus,

(A(φ(x)), φ(h(x))) = τ(A(φ(x))).

Since φ sends h(x) to φ(h(x)) but sends bdSS+ to bdSS−, it follows thatφ sends the component Ho

x of E−h(x) to the component (E−HA(φ(x)))o

of E− φ(h(x)) for each fixed point x of an element of Γ. We obtain

φ(Hox) = (E−HA(φ(x)))

o for x ∈ Λ.

Since the domain D is determined by h(x) for x ∈ Λ, we obtain φ(D) =D′ and φ(D) = D′.

The last statement follows by Lemma 9.4.

Remark 9.6. These domains are the invariant domains found by Mess[43] for the closed surface groups. See Barbot [3] and Bonsante [8] fordetails of Mess’s work around this. Also, Barbot had shown the factthat the supporting hyperplanes contain all nonwandering geodesics.

10. Geometric finiteness

We define geometric finiteness and show that the Margulis space-times are geometrically finite. The geometrical finiteness here of courseis already discovered by Goldman, Labourie, and Margulis [35], [34],and we only restate it in this form.

We start with

56 CHOI AND GOLDMAN

Theorem 10.1 (Scott [46]). If M is an orientable 3-manifold andπ1(M) is finitely generated, then there exists a compact submanifold Nof M such that the inclusion homomorphism induces an isomorphismπ1(N)→ π(M).

If M is aspherical, as in Margulis spacetimes, then we can construct acompact submanifold so that the inclusion map N →M is a homotopyequivalence. We say that a compact submanifold with properties of Nis a core of M . For E/Γ with a free group Γ of a finite rank, there existsa core.

Definition 10.2 (Geometric finitness). Let Γ be the orientation-preservingLorentzian finitely generated isometry group acting on the Lorenzianspace E freely and properly discontinuously. Let E/Γ be given an arbi-trary complete path-metric δ and a core M0. Let R denote the set ofall nonwandering geodesics in E/Γ. E/Γ is geometrically finite if⋃

c∈R

c ⊂ N δC(M0)

for a C-δ-neighborhood of M0 for a constant C > 0.

In other words, for each pair of a nonwandering geodesic c and apoint x on c in E/Γ, x is of a uniformly bounded δ-distance from M0;that is, δ(x,M0) ≤ C for a constant C independent of x and c.

Actually, since M0 is compact, we can replace M0 by a base point x0

in the above definition.

Lemma 10.3. Fix an arbitrary point x0 and an arbitrary Euclideanmetric dE on E considered as an affine space. A Margulis spacetimeE/Γ is geometrically finite if and only if there exists a constant C ′ > 0such that for every pair of a geodesic l mapping to a nonwanderinggeodesic in E/Γ and a point y on l in E, there exists γ ∈ Γ so thatdE(x0, γ(y)) < C ′.

Proof. Let C > 0 be the uniform bound for the path-metric δ of E/Γused in Definition 10.2. A δ-ball of radius C in E maps to a boundedball in E in the Euclidean metric dE since it is compact. The Euclideandiameters of the δ-balls of radii equal to C with centers in a compactset K are bounded above by a constant C ′ > 0.

Let l be a geodesic with y ∈ l in the premise. Suppose that l mapsto a nonwandering geodesic c and y to a point x ∈ c. Then x is of theδ-distance bounded above by C from a point x′ ∈ M0 by definition ofthe geometric finiteness. Let F be the compact fundamental domainof M0 in E. Assume without loss of generality that x0 ∈ F . Choose


γ ∈ Γ so that γ(y) is of the δ-distance ≤ C from x′ ∈ F . Thus, γ(y) isof the Euclidean distance ≤ C ′ from x0.

Conversely, a Euclidean ball with center x0 of radius C ′ has a δ-diameter uniformly bounded above. The proof of the converse is verysimilar to the above.

Theorem 10.4. A Margulis spacetime E/Γ without parabolics is geo-metrically finite.

Proof. Assume first L(Γ) ⊂ SO(2, 1)o. Let x0 be a base point in E.

Consider the compact fundamental domain F of UrecS+ under the decktransformation group to UrecΣ+ which covers a closed subset of thecompact set UΣ′′+. Then πE N(F ) gives us a set A of geodesics ofbounded distances from x0 since it is an image of a compact set: thatis, dS3(x0, l)|l ∈ A < C for a constant C.

Let u denote the unit vector at t in E tangent to a geodesic l mappingto a nonwandering one in E/Γ. Then (t, u) corresponds to (t′, u′) for apoint t′ ∈ S+ and u′ on UrecS+ via πE N by Proposition 6.2. SinceF is a fundamental domain, (γ(t′), Dγ(u′)) is in the bounded set F fora deck transformation γ and the differential Dγ. Thus, (γ(g), Dγ(u))

is in the image of πE N(F ), a compact subset. By Lemma 10.3, itfollows that E/Γ is geometrically finite.

Assume now that L(Γ) is not a subset of SO(2, 1)o. The full generalcase follows by Lemma 9.4.

Theorem 10.5. There exists a core in a Margulis spacetime containingall nonwandering spacelike geodesics.

Proof. Let Γ be the finitely generated free group of rank ≥ 2 actingfreely and properly discontinuously on E. We construct Σ as above.

First, assume L(Γ) is a subset of SO(2, 1)o.Let M denote the compact handlebody (E∪ Σ)/Γ so that M o is the

Margulis spacetime E/Γ. We denote by M the space E∪ Σ. Recall thedomains D and D′ from the proof of Corollary 9.5. Then consideringthe closure D = ClM(D) of D in M , it follows that Γ acts properly

discontinuously on D as well.The topological boundary bdMD in M is a closed invariant subset

of E and is a connected properly imbedded 2-dimensional submanifoldinvariant under Γ. Since bdMD equals bdD −D(Λ), bdMD is homeo-

morphic to a 2-cell. However, under the projection p : M →M , it goesto a compact closed connected properly imbedded surface Σ1 boundingthe image of D. To explain more, M1 := D/Γ is imbedded in M and

58 CHOI AND GOLDMAN

is homotopy equivalent to M and the topological boundary bdMM1 inM is the compact surface Σ1.

Since Σ1 is homotopic to the closure S1 of D(Λ)/Γ in Σ sharingboundary components in M1 and ∂M1 = Σ1 ∪ S1, it follows that M1 ishomeomorphic to Σ1 × I for an interval I by Theorem 10.2 of [38].

Notice also

∂Σ1 = ∂S1 =b⋃i=1

ci

for the number of boundary components b of S1 and the boundarycomponents ci of Σ1.

We can do similarly for the closure D− := ClM(A(D)) and we obtain

M2 := D−/Γ imbedded into M disjoint from M1. Similarly M2 ishomeomorphic to Σ2 × I for a surface Σ2 = bdMM2 which is alsohomotopy equivalent to M .

Notice also

∂Σ2 =b⋃i=1

c′i

for the boundary components c′i of Σ2.Let intM1 and intM2 denote the topological interiors. Since we are

removing the product of I with surfaces, it follows that M − intM1 −intM2 is a compact 3-manifold homeomorphic to M bounded by Σ1 ∪Σ2 ∪B1 ∪ · · · ∪Bg. (Recall Bi from Section 9.2.)

By geometric finiteness, every nonwandering geodesic is of a uni-formly bounded δ-distance from a base point x0 in M . Therefore,we can isotopy all Bi to imbedded annuli in M o so that the resultingimbedded annuli B′i are all disjoint from closed spacelike geodesics inM o and ∂B′i ⊂ Σ1 ∪Σ2. Therefore, Σ1 and Σ2 and B′1, . . . , B

′g bound a

compact submanifold M containing all closed spacelike geodesics.Assume now that L(Γ) is not a subset of SO(2, 1)o. The full general

case follows since a coset representative φ ∈ Γ − Γ′ induces a diffeo-morphism of M sending M1 to M2 and vice versa and by Lemma 9.4.We can also choose Bi equivariantly.

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Department of Mathematical Sciences, KAIST, Daejeon 305-701, SouthKorea,

E-mail address: [email protected]

Department of Mathematics, University of Maryland, College Park,MD 20742-4015, USA

E-mail address: [email protected]

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