MARGULIS SPACETIMES VIA THE ARC COMPLEX - … · margulis spacetimes via the arc complex preliminary version (january 2014) | check back soon for an updated draft. jeffrey danciger,

MARGULIS SPACETIMES VIA THE ARC COMPLEX

PRELIMINARY VERSION (JANUARY 2014) — CHECK BACK SOON FOR AN

UPDATED DRAFT.

JEFFREY DANCIGER, FRANCOIS GUERITAUD, AND FANNY KASSEL

Abstract. We study strip deformations of convex cocompact hyper-bolic surfaces, defined by inserting hyperbolic strips along a collection ofdisjoint geodesic arcs properly embedded in the surface. We prove thatany (infinitesimal) deformation of the surface that uniformly lengthensall closed geodesics can be realized as an (infinitesimal) strip deforma-tion, in an essentially unique way. This gives a parameterization, bythe arc complex, of the moduli space of Margulis spacetimes with fixedconvex cocompact linear holonomy. As an application, we give a newproof of the tameness of such Margulis spacetimes by establishing theCrooked Plane Conjecture, which states that any Margulis spacetime ad-mits a fundamental domain bounded by piecewise linear surfaces calledcrooked planes. We also prove analogous results for complete anti-deSitter 3-manifolds.

1. Introduction

The understanding of moduli spaces using simple combinatorial models isa major theme in geometry. While coarse models, like the curve complex orpants complex, are used to great effect in the study of the various metricsand compactifications of Teichmuller spaces (see e.g. [MM, BM]), parame-terizations and/or cellulations can provide insight at both macroscopic andmicroscopic scales. One prominent example is Penner’s cell decompositionof the decorated Teichmuller space of a punctured surface [P]. This pa-per gives a parameterization, similar and related to Penner’s, of the modulispace of certain Lorentzian 3-manifolds called Margulis spacetimes.

A Margulis spacetime is a quotient of the 3-dimensional Minkowski spaceR2,1 by a free group Γ acting properly discontinuously by isometries. Thefirst examples were constructed by Margulis [Ma1, Ma2] in 1983, as counter-examples to Milnor’s suggestion [Mi] to remove the cocompactness assump-tion in the Auslander conjecture [Au]. Since then many authors, most promi-nently Charette, Drumm, Goldman, Labourie, and Margulis, have studiedtheir geometry, topology, and deformation theory. Any Margulis spacetimeis determined by a noncompact hyperbolic surface S, with π1(S) = Γ, and

J.D. is partially supported by the National Science Foundation under the grant DMS1103939. F.G. and F.K. are partially supported by the Agence Nationale de la Rechercheunder the grants DiscGroup (ANR-11-BS01-013) and ETTT (ANR-09-BLAN-0116-01),and through the Labex CEMPI (ANR-11-LABX-0007-01).

1

MARGULIS SPACETIMES VIA THE ARC COMPLEX 2

an infinitesimal deformation of S called a proper deformation. The subsetof proper deformations forms a symmetric cone, which we call the admissi-ble cone, in the tangent space to the Fricke space F of complete hyperbolicstructures on S. In the case that S is convex cocompact, seminal workof Goldman–Labourie–Margulis [GLM1] shows that the admissible cone isopen with two convex components, consisting of the infinitesimal deforma-tions of S that uniformly expand or uniformly contract the marked lengthspectrum of S.

In this paper we study a simple geometric construction, called a stripdeformation, which produces uniformly expanding deformations of S: it isdefined by cutting S along finitely many disjoint, properly embedded ge-odesic arcs, and then gluing in a hyperbolic strip, i.e. the region betweentwo ultraparallel geodesics in H2, at each arc. An infinitesimal strip de-formation (Definition 1.3) is the derivative of a path of strip deformationsalong some fixed arcs as the widths of the strips decrease linearly to zero.It turns out that, as soon as the supporting arcs decompose the surfaceinto disks, an infinitesimal strip deformation lengthens all closed geodesicsof S uniformly (this also holds for noninfinitesimal deformations, as was ob-served by Thurston [T] and proved by Papadopoulos and Theret [PT]); thusit is a proper deformation. Our main result (Theorem 1.4) states that allproper deformations of S can be realized as infinitesimal strip deformations,in an essentially unique way: after fixing some choices about the geometryof the strips, the map from the complex of arc systems on S to the projec-tivization of the admissible cone, taking any weighted system of arcs to thecorresponding infinitesimal strip deformation, is a homeomorphism.

We note that infinitesimal strip deformations are also used by Goldman–Labourie–Margulis–Minksy in [GLMM]. They construct modified infinites-imal strip deformations along arcs which accumulate on a geodesic lamina-tion in order to describe infinitesimal deformations of a surface for which alllengths increase, but fail to increase uniformly.

As an application of our main theorem, we give a new proof of the tame-ness of Margulis spacetimes, under the assumption that the associated hy-perbolic surface is convex cocompact. This result was recently established byChoi–Goldman [ChoG] and by the authors [DGK1], independently. Here weactually prove the stronger result, dubbed the Crooked Plane Conjecture byDrumm–Goldman [DG1], that any Margulis spacetime admits a fundamen-tal domain bounded by crooked planes, piecewise linear surfaces introducedby Drumm [D]. In the case that the free group Γ has rank two, the con-jecture was verified by Charette–Drumm–Goldman [CDG3]. In particular,when the surface S is a one-holed torus, they find a tiling of the admissiblecone according to which triples of isotopy classes of crooked planes embeddisjointly, a picture which is generalized, in spirit, by our parameterizationby strip deformations. We show that a strip deformation encodes precisedirections for building fundamental domains in R2,1 bounded by crookedplanes (Section 7.4).


We also prove analogous results in the related setting of complete anti-deSitter 3-manifolds, but defer a description of these results until later in theintroduction (Section 1.4). We now state precisely our main results.

1.1. Margulis spacetimes. The 3-dimensional Minkowski space R2,1 is theaffine space R3 endowed with the parallel Lorentzian structure induced by aquadratic form of signature (2, 1); its isometry group is O(2, 1)nR3, actingaffinely. Let G be the group PGL2(R), acting on the real hyperbolic planeH2 by isometries in the usual way, and on the Lie algebra g = pgl2(R) bythe adjoint action. We shall identify R2,1 with the Lie algebra g endowedwith the Lorentzian structure induced by the Killing form. The group oforientation-preserving isometries of R2,1 identifies with Gn g.

By [FG] and [Me], if a nonsolvable group Γ acts properly discontinuouslyand freely by isometries on R2,1, then Γ is a free group and its action on R2,1

is orientation-preserving (see e.g. [Ab]) and induces an embedding of Γ intoGn g with image

(1.1) Γρ,u = (ρ(γ), u(γ)) | γ ∈ Γ ⊂ Gn g

where ρ ∈ Hom(Γ, G) is a discrete and injective representation and u : Γ→ ga ρ-cocycle, i.e. u(γ1γ2) = u(γ1) + Ad(ρ(γ1))u(γ2) for all γ1, γ2 ∈ Γ. Bydefinition, a Margulis spacetime is a manifold M = Γρ,u\R2,1 determinedby such a proper group action. Properness is invariant under conjugationby G n g. We shall consider conjugate proper actions to be equivalent; inother words, we shall consider Margulis spacetimes to be equivalent if thereexists a marked isometry between them. In particular, we will be interestedin holonomies ρ up to conjugacy, i.e. as classes in Hom(Γ, G)/G, and inρ-cocycles u up to addition of a coboundary, i.e. as classes in H1

ρ (Γ, g) =

H1(Γ, gAd ρ).Let Γρ,u\R2,1 be a Margulis spacetime. The representation ρ is the holo-

nomy representation of a noncompact hyperbolic surface S = ρ(Γ)\H2, anda ρ-cocycle u can be interpreted as an infinitesimal deformation of this holo-nomy, obtained as the derivative u(γ) = d

dt |t=0 ρt(γ)ρ(γ)−1 of some path of

representations ρt(γ) = etu(γ)+o(t)ρ(γ) (see [DGK1, § 2.3] for instance). Thusthe deformation space of Margulis spacetimes projects to the Fricke spaceof noncompact hyperbolic surfaces; describing the fiber above S amounts toidentifying the proper deformations u of ρ, i.e. the infinitesimal deformationsu of ρ for which the group Γρ,u acts properly discontinuously on R2,1.

A properness criterion was given by Goldman–Labourie–Margulis [GLM1]:suitably interpreted [GM], it states that for convex cocompact ρ ∈ Hom(Γ, G)the group Γρ,u acts properly discontinuously on R2,1 if and only if the infin-itesimal deformation u “uniformly lengthens all closed geodesics”, i.e.

(1.2) infγ∈Γre

dλγ(u)

λγ(ρ)> 0,


or “uniformly contracts all closed geodesics”, i.e. (1.2) holds for −u insteadof u. Here λγ : Hom(Γ, G) → R+ is the function (2.1) assigning to anyrepresentation ρ the hyperbolic translation length of ρ(γ). That Γ is convexcocompact means that Γ is finitely generated and that the discrete groupρ(Γ) ⊂ G does not contain any parabolic element; equivalently, S = ρ(Γ)\H2

is the union of a minimal compact convex subset (called the convex core) andof finitely many ends of infinite volume (called the funnels). In [DGK1] wegave a new proof of the Goldman–Labourie–Margulis criterion, as well as an-other equivalent properness criterion in terms of expanding (or contracting)equivariant vector fields on H2. These criteria are to be extended to arbi-trary injective and discrete ρ (for finitely generated Γ) in [GLM2, DGK2],allowing ρ(Γ) to have parabolic elements.

We introduce the following terminology.

Definition 1.1. The positive admissible cone in H1ρ (Γ, g) is the subset of

classes of ρ-cocycles u satisfying (1.2). The admissible cone is the union ofthe positive admissible cone and of its opposite. The projectivization of theadmissible cone, a subset of P(H1

ρ (Γ, g)), will be denoted adm(ρ).

The positive admissible cone is an open, convex cone in the finite-dimensionalvector space H1

ρ (Γ, g).We now describe the fundamental objects of the paper, namely strip de-

formations, which will be used to parameterize adm(ρ) (Theorem 1.4).

1.2. The arc complex and strip deformations. Let S be a convex co-compact hyperbolic surface with holonomy representation ρ and fundamen-tal group Γ = π1(S). We call arc of S any nontrivial isotopy class of em-bedded lines in S for which each end exits in a funnel. The following notionwas first introduced by Thurston [T, proof of Lem. 3.4].

Definition 1.2. Let α1, . . . , αk+1 be pairwise disjoint arcs of S. A stripdeformation of S is a hyperbolic surface S′ that is obtained from S asfollows: for every 1 ≤ i ≤ k + 1, cut along a geodesic representative αiof αi and glue in (without any shearing) a strip of H2, bounded by twoultraparallel geodesics. We shall also say that the holonomy of S′ (definedup to conjugation) is a strip deformation of the holonomy ρ of S.

We call the point pαi ∈ αi where the glued strip has smallest width the waistof the strip.We shall use the infinitesimal version of this construction:

Definition 1.3. An infinitesimal strip deformation of S is the class inH1ρ (Γ, g) of a ρ-cocycle u : Γ → g obtained as the derivative at t = 0 of

a path t 7→ ρt ∈ Hom(Γ, G) of strip deformations of ρ along fixed geodesicsα1, . . . , αk+1, with fixed waists, and such that the widths of the strips, mea-sured at the waists, are of the form mit for some fixed m1, . . . ,mk+1 > 0.

In order to parameterize the admissible cone by strip deformations, wemake certain choices. For each arc α of S, fix:


pα pα

pαα

α

α

Figure 1. A strip deformation along one arc of a four-holed sphere

• a geodesic representative α of α,• a point pα ∈ α (the waist),• a positive number mα > 0 (the width).

We require that the α intersect minimally, meaning that the representativesα1 and α2 of two arcs α1 and α2 always have smallest possible intersectionnumber (including ideal intersection points). This can be achieved by choos-ing the representatives α to intersect the convex core boundary orthogonally,but we do not require this.

For any arc α of S, define f(α) ∈ H1ρ (Γ, g) to be the infinitesimal strip

deformation of ρ along α with waist pα and width mα. Recall that thearc complex X of S is the simplicial complex whose vertices are the arcsof S, with one k-dimensional simplex for each collection of k + 1 pairwisehomotopically disjoint arcs. The map α 7→ f(α) extends by barycentricinterpolation to a map f : X → H1

ρ (Γ, g). By postcomposing with the

projectivization map H1ρ (Γ, g)→ P(H1

ρ (Γ, g)), we obtain a map

f : X −→ P(H1ρ (Γ, g)

).

Let X be the subset of X obtained by removing all faces corresponding tocollections of arcs that do not subdivide the surface S into disks. No vertexof X is in X, but the interior of any top-dimensional cell is (see Section 6 forexamples). By work of Penner [P] on the decorated Teichmuller space, X ishomeomorphic to an open ball of dimension 3|χ| − 1, where χ is the Eulercharacteristic of S. We prove that any point of the positive admissible coneis realized as an infinitesimal strip deformation, in a unique way given ourchoices of (α, pα,mα):

Theorem 1.4. The map f restricts to a homeomorphism between X andthe projectivized admissible cone adm(ρ).

It is natural to wonder about the image of f in H1ρ (Γ, g), before projec-

tivization. Since adm(ρ) = f(X) is convex, it seems reasonable to expectthat f(X) should appear as the boundary of a convex object in H1

ρ (Γ, g):

Conjecture 1.5. There exists a choice of minimally intersecting geodesicrepresentatives α and waists pα such that if all the widths mα are 1, thenf(X) is a convex hypersurface in H1

ρ (Γ, g).


1.3. Fundamental domains for Margulis spacetimes. In 1992, Drumm[D] introduced piecewise linear surfaces in Minkowski space R2,1 called crook-ed planes (see [ChaG]). The Crooked Plane Conjecture of Drumm andGoldman states that any Margulis spacetime admits a fundamental domainin R2,1 bounded by finitely many crooked planes. This conjecture was provedby Charette–Drumm–Goldman [CDG1, CDG3] in the special case that thefundamental group is a free group of rank two. Here we give a generalproof of the Crooked Plane Conjecture when the linear holonomy is convexcocompact.

Theorem 1.6. Any discrete subgroup of SO(2, 1) nR3 acting properly dis-continuously and freely on R2,1, with convex cocompact linear part, admitsa fundamental domain in R2,1 bounded by finitely many crooked planes.

This is an easy consequence of Theorem 1.4: the idea is to interpret stripdeformations as motions of crooked planes making them disjoint in R2,1 (seeSection 7.4). Theorem 1.6 provides a new proof of the tameness of Margulisspacetimes with convex cocompact linear holonomy, independent from theoriginal proofs given in [ChoG, DGK1].

1.4. Strip deformations and anti-de Sitter 3-manifolds. In [DGK1],we demonstrated that, in a precise sense, Margulis spacetimes behave like“infinitesimal analogues” of complete AdS manifolds, which are quotientsof the negatively-curved anti-de Sitter space AdS3. Following this point ofview further, we now derive analogues of Theorems 1.4 and 1.6 for AdSmanifolds.

The anti-de Sitter space AdS3 = PO(2, 2)/O(2, 1) is a model space forLorentzian manifolds of constant negative curvature. It can be realized asthe set of negative points in P3(R) with respect to a quadratic form ofsignature (2, 2); its isometry group is PO(2, 2). Equivalently, AdS3 can berealized as the group G0 = PSL2(R) < G endowed with the biinvariantLorentzian structure induced by the Killing form of g = pgl2(R) = psl2(R);the group of orientation-preserving isometries then identifies with

(G×G)+ := (g1, g2) ∈ G×G | g1g2 ∈ G0,acting on G0 by right and left multiplication: (g1, g2) · g = g2gg

−11 .

By [KR], any torsion-free discrete subgroup of (G × G)+ acting prop-erly discontinuously on AdS3 is, up to switching the two factors of G × G,of the form

Γρ,j = (ρ(γ), j(γ)) | γ ∈ Γ ⊂ G×Gwhere Γ is a discrete group and ρ, j ∈ Hom(Γ, G) two representations withj injective and discrete. Suppose that Γ is finitely generated. By [Ka, GK],a necessary and sufficient condition for the action of Γρ,j on AdS3 to beproperly discontinuous is that (up to switching the two factors) j be injectiveand discrete and ρ be “uniformly contracting” with respect to j, in the sensethat there exists a (j, ρ)-equivariant Lipschitz map H2 → H2 with Lipschitz


constant < 1, or equivalently that

(1.3) supγ∈Γre

λγ(ρ)

λγ(j)< 1,

where λγ : Hom(Γ, G)→ R+ again denotes the hyperbolic translation lengthof γ. If Γ is a cocompact surface group, and both j, ρ are injective anddiscrete, then (1.3) is never satisfied [T].

Suppose that ρ is convex cocompact, of infinite covolume, and let S bethe hyperbolic surface ρ(Γ)\H2, with holonomy ρ. We denote by F the set ofconjugacy classes of convex cocompact holonomies of hyperbolic structureson the topological surface underlying S (the “convex cococompact Frickespace” of S), and by Adm+(ρ) ⊂ F the subset corresponding to convex co-compact representations j ∈ Hom(Γ, G) that are “uniformly longer” than ρ,namely that satisfy (1.3). As in Section 1.2, for each arc α of S we fixa geodesic representative α of α, a point pα ∈ α, and a positive numbermα > 0, and we require that the α intersect minimally. Let F (α) ∈ F bethe class of the strip deformation of ρ along α with waist pα and smalleststrip width mα. Since the vertices of a cell of the arc complex X correspondto disjoint arcs, the cut-and-paste operations along them do not interfereand the map α 7→ F (α) naturally extends to a map F : CX → F, whereCX is the abstract cone over the arc complex X. Let CX ⊂ CX be theabstract open cone over X. We prove the following “macroscopic” versionof Theorem 1.4.

Theorem 1.7. For convex cocompact ρ, the map F restricts to a homeo-morphism between CX and Adm+(ρ).

In other words, any “uniformly lengthening” deformation of ρ can be real-ized as a strip deformation, and the realization is unique, once the geodesicrepresentatives α, waists pα, and widths mα are fixed for all arcs α.

By analogy to the Minkowski setting, it is natural to ask whether allfree, properly discontinuous actions on AdS3 admit fundamental domainsbounded by nice polyhedral surfaces. In Section 8, we introduce piecewisegeodesic surfaces in AdS3 that we call AdS crooked planes, and prove:

Theorem 1.8. Let j, ρ ∈ F be the holonomy representations of two convexcocompact hyperbolic structures on a fixed hyperbolic surface S and assumethat Γρ,j acts properly discontinuously on AdS3. Then Γρ,j admits a funda-mental domain bounded by finitely many AdS crooked planes.

Theorem 1.8 provides a new proof of the tameness (obtained in [DGK1])for AdS manifolds, in this special case.

Note that the situation is very different when Γ is the fundamental groupof a compact surface: as mentioned above, in this case j is Fuchsian andρ is necessarily non-Fuchsian [T]. By [GKW] or [DT], the subset Adm+(ρ)of the Fricke space (here Teichmuller space) consisting of representations j“uniformly longer” than ρ is always nonempty; it would be interesting to


obtain a parameterization of Adm+(ρ) by some simple combinatorial objectin this situation as well.

1.5. Organization of the paper. In Section 2 we prove some technicalestimates for infinitesimal strip deformations. In Section 3 we reduce theproofs of Theorems 1.4 and 1.7 to Claim 3.2, about the behavior of themap f at faces of codimension ≤ 2. This claim is proved in Section 5.Before that, in Section 4, we introduce some formalism that is also usefullater in the paper. In Section 6 we then give some basic examples of thetiling of the admissible cone produced by Theorem 1.4. Finally, Sections7 and 8 are devoted to the proofs of Theorems 1.6 (the Crooked PlaneConjecture) and 1.8 about the construction of fundamental domains usingstrip deformations.

Acknowledgments. We would like to thank Thierry Barbot, Virginie Cha-rette, Todd Drumm, and Bill Goldman for interesting discussions related tothis work, as well as Francois Labourie and Yair Minsky for igniting remarks.We are grateful to the Institut Henri Poincare in Paris and to the Centrede Recherches Mathematiques in Montreal for giving us the opportunity towork together in stimulating environments. Finally, we acknowledge supportfrom the GEAR Network, funded by the National Science Foundation undergrant numbers DMS 1107452, 1107263, and 1107367 (“RNMS: GEometricstructures And Representation varieties”).

2. Metric estimates for (infinitesimal) strip deformations

Let S be a convex cocompact hyperbolic surface of infinite volume, withfundamental group Γ = π1(S) and holonomy representation ρ ∈ Hom(Γ, G).Let F be the convex cocompact Fricke space of S, by which we mean thespace of conjugacy classes of holonomy representations of convex cocompacthyperbolic structures on S. The tangent space T[ρ]F identifies with the space

H1ρ (Γ, g) of cohomology classes of ρ-cocycles.For any γ ∈ Γ and any τ ∈ Hom(Γ, G) we set

(2.1) λγ(τ) := infp∈H2

d(p, τ(γ) · p),

where d is the hyperbolic metric on H2: this is the translation length ofτ(γ) if τ(γ) ∈ G is hyperbolic, and 0 otherwise. In particular, this defines afunction λγ : F→ R+, whose differential is denoted by dλγ : TF→ R.

As in Section 1.2, for each properly embedded arc α in S we fix a geodesicrepresentative α of α, a point pα ∈ α, and a positive number mα > 0,such that the geodesic representatives α have minimal intersection numbers(in H2 ∪ ∂∞H2). One such system of geodesic representatives is given bythe geodesic arcs which are orthogonal to the convex core boundary. Anyother such system is obtained from it by pushing the endpoints at infinity ofthe arcs forward by an isotopy of the circles at infinity of the funnels, whichyields the following.


Remark 2.1. The space of all choices of geodesic representatives α andwaists pα ∈ α as above is connected.

When the context is clear, we will sometimes refer to the geodesic repre-sentatives α simply as arcs.

2.1. Variation of length of geodesics under strip deformations. LetX be the arc complex of S. For x ∈ X, we denote by |x| ⊂ S the supportof x, i.e. the union of the geodesic arcs α corresponding to the vertices ofthe smallest cell of X containing x. Let ϑx : |x| → R+ be the strip widthfunction mapping any p ∈ α ⊂ |x| to

(2.2) ϑx(p) := wα cosh d(p, pα),

where wα is the width (as measured at the narrowest point) of the strip tobe inserted along α, and the distance d(·, ·) is implicitly measured along α.Note that wα is mα times the weight of α in the barycentric expression forx ∈ X, and that ϑx(p) is the length of the path crossing the strip at p atconstant distance from the waist segment. For p ∈ H2, let ]p denote themeasure of angles at p, valued in [0, π]. Then the following formula holds.

Observation 2.2. For any γ ∈ Γ r e and any x ∈ X,

(2.3) dλγ(f(x)

)=

∑p∈γ∩|x|

ϑx(p) sin]p(|x|, γ),

where γ is the geodesic representative of γ on S.

Formula (2.3) is analogous to the cosine formula expressing the effect ofan earthquake on the length of a closed geodesic (see [Ke] for example), andis proved similarly. The difference is that the angle ]p(|x|, γ) in the formulafor earthquakes is replaced by its complementary to π/2 in the formulafor strip deformations, changing the cosine into a sine. In general, stripdeformations of ρ should be thought of as analogues of earthquakes, whereinstead of sliding against itself, the surface is pushed apart in a directionorthogonal to each arc of the support. In (2.3), unlike in the formula forearthquakes, the contribution of each intersection point p depends not onlyon the angle, but also (via ϑx) on p itself.

Proof of Observation 2.2. By linearity, it is sufficient to prove the formulawhen x is a vertex α of the arc complex X, and the infinitesimal width wαat the waist is unit. For t ∈ R, we set

at :=

(et/2 0

0 e−t/2

), bt :=

(cosh t

2 sinh t2

sinh t2 cosh t

2

), rt :=

(cos t

2 sin t2

− sin t2 cos t

2

).

Up to conjugation we may assume that ρ(γ) = aλγ(ρ). Suppose the geodesicloop γ crosses the arc α at points q1, . . . , qk, in this order. For 1 ≤ i ≤ k,let ì be the distance along γ between qi−1 and qi (with the convention thatq0 = qk), let di be the signed distance along α from qi to the waist pα, and


let θi := ]qi(γ, α). Then up to conjugation

F (tx)(γ) = a`1(rθ1ad1bta−d1r−θ1) a`2(. . . ) a`k(rθkadkbta−dkr−θk) ∈ G.Note that, ignoring nondiagonal entries,

d

dt

∣∣∣∣t=0

(rθiadibta−dir−θi

)=

sin θi · cosh di2

(1 ∗∗ −1

).

Therefore, setting ` := `1 + · · ·+ `k = λγ(ρ), we have

F (tx)(γ) = ρ(γ) + t

(k∑i=1

sin θi·cosh di2 a`1+···+ì

(1 ∗∗ −1

)aì+1+···+`k

)+ o(t)

=

(e`/2 0

0 e−`/2

)+t

2

(k∑i=1

sin θi cosh di

(e`/2 ∗∗ −e−`/2

))+ o(t).

By differentiating the formula λ(g) = 2 arccosh(tr(g)/2) for g ∈ G (whereλ(g) is the translation length of g in H2), we find

dλγ(f(x)

)=

k∑i=1

sin θi cosh di =

k∑i=1

ϑx(qi) sin θi.

2.2. Angles at the boundary of the convex core. A hyperideal trian-gulation ∆ of S corresponds to a top-dimensional face of the arc complex X,which is the span of 3|χ| arcs dividing the surface into 2|χ| hyperideal tri-angles. (Here χ ∈ −N is the Euler characteristic of S.)

Proposition 2.3. There exists θ0 > 0 such that all arcs α intersect theboundary of the convex core of S at an angle ≥ θ0. Moreover, θ0 can betaken to depend continuously on ρ and on the choices of geodesic arcs of oneparticular hyperideal triangulation ∆.

Proof. This is a consequence of the hypothesis that the geodesic arcs αintersect minimally. Indeed, consider one boundary component η of S, andchoose one arc β of ∆ that exits η (at least at one end). In the universal

cover H2, a lift η of η is intersected by lifts (βi)i∈Z of β, all disjoint. Each βiescapes through two boundary components of the lift of the convex core (ηand another one): let β∗i denote the arc perpendicular to these two boundarycomponents. In particular, the β∗i are lifts of the arc β∗ of S which is in the

same class as β, but intersects the convex core boundary at right angles.Consider another arc α of S which is different from β, but also crosses η.

Let α∗ denote the representative of α which is orthogonal to the convex coreboundary. In H2, a lift of α∗ that intersects η stays between β∗i and β∗i+1, forsome i ∈ Z. By minimality of the intersection numbers, the correspondinglift of α stays between βi and βi+1. As a consequence, the angle at which αintersects η can be bounded away from 0, independently of α. We concludeby repeating for all boundary components η.


2.3. The unit-peripheral normalization. Let ∂S denote the boundaryof the convex core of (S, j). By (2.3), choosing all widths mα so that

(2.4)∑

p∈|x|∩∂S

ϑx(p) sin]p(|x|, ∂S) = 1

for all x ∈ X (or equivalently, for all vertices x of X) corresponds to askingall infinitesimal strip deformations associated with the choices (α, pα,mα)to lengthen ∂S at unit rate. In particular, f(X) is contained in some affinehyperplane of T[ρ]F = H1

ρ (Γ, g), which justifies the following terminology.

Definition 2.4. We call (2.4) the unit-peripheral normalization.

Due to Proposition 2.3, in the unit-peripheral normalisation,

(2.5) ϑx(p) ≤ 1

sin θ0

for all x ∈ X and p in the boundary of the convex core of (S, ρ); by (2.2)and convexity of cosh, the same bound holds in fact for all p in the convexcore.

Using (2.3), we bound the variation of length of closed geodesics underinfinitesimal strip deformations.

Proposition 2.5. In the unit-peripheral normalization, there exists K > 0(depending on ρ and on the chosen geodesic representatives α) such that forall γ ∈ Γ r e and x ∈ X,∑

p∈γ∩|x|

ϑx(p) ≤ KRγ,x λγ(ρ),

where γ is the closed geodesic of S corresponding to γ and

Rγ,x := maxp∈γ∩|x|

]p(γ, |x|).

Proof. Fix γ ∈ Γre. Any lift γ of γ to H2 intersects the preimage of |x| atpoints pii∈Z, naturally ordered along γ, so that the deck transformationdetermined by γ takes each pi to pi+m. Let ì ⊂ H2 be the lifted arc of |x|that contains pi, and let ϑx be the lift of the function ϑx to H2. Recall thatthe support |x| consists of at most 3N arcs, where we set N := |χ|. It isenough to find K ′ ≥ 0 such that

ϑx(pi) ≤ K ′Rγ,x d(pi, pi+6N ) ;

indeed the result will follow (with K = 6NK ′) by adding up for 1 ≤ i ≤ m.Assume first that d(pi, pi+6N ) ≥ 1. For any ε > 0, there exists κ > 0 such

that if Rγ,x is smaller than some value R0 > 0, then ì stays ε-close to γ forat least

| logRγ,x| − κunits of length to the left and right of pi. If ε is small enough, then in par-ticular ì cannot exit the lift Ω of the convex core on this interval (otherwiseγ would exit too, by Proposition 2.3). But on this interval, the maximum


of ϑx|ì is at least 12e| logRγ,x|−κ ϑx(pi), by definition of ϑx. Using (2.5), it

follows that ϑx(pi) ≤ Rγ,x2eκ

sin θ0, hence the desired bound. If Rγ,x ≥ R0,

then Rγ,x d(pi, pi+6N ) ≥ R0 whereas ϑx(pi) is bounded from above by (2.5),hence again the desired bound.

Assume now that d(pi, pi+6N ) ≤ 1. The distance from the point pi ∈ ìto the line ì+6N is at most δ := 2Rγ,x d(pi, pi+6N ). Suppose that ì andì+6N exit different boundary components of Ω (on both sides of γ). Forany ε > 0, there exists κ > 0 such that if δ is smaller than some δ0 > 0,then ì stays ε-close to ì+6N for

| log δ| − κunits of length to the right and left of pi. If ε is small enough, then inparticular ì cannot exit Ω on this interval (otherwise ì+6N would exit

too, by Proposition 2.3). But on this interval, the maximum of ϑx|ì is

at least 12e| log δ|−κ ϑx(pi) by definition of ϑx. Using (2.5), it follows that

ϑx(pi) ≤ δ 2eκ

sin θ0, hence the desired bound. If δ ≥ δ0, just use the upper

bound (2.5) on ϑx(pi).The remaining case is when ì and ì+6N exit the same boundary com-

ponent η of Ω. Since |x| has at most 6N half-arcs exiting any boundarycomponent of S, it follows that ì and `j are related by a deck transforma-tion stabilizing η, for some i < j ≤ i+ 6N . By Proposition 2.3, this impliesthat the shortest distance from ì to `j is bounded from below; thereforeso is d(pi, pi+6N ). We can then apply the same argument as in the cased(pi, pi+6N ) ≥ 1.

2.4. The map f is bounded. Equation (2.3) and Proposition 2.5 implythat in the unit-peripheral normalization (2.4),

(2.6)dλγ

(f(x)

)λγ(ρ)

≤ K(

maxp∈γ∩|x|

]p(γ, |x|))2.

The square on the right-hand side can be interpreted by saying that smallintersection angles in γ ∩ |x| weaken the effect of the strip deformation f(x)on the length of γ, in two different ways: first, by spreading out the inter-section points along γ (Proposition 2.5); second, by lessening the effect ofeach (Formula (2.3)).

Proposition 2.6. In the unit-peripheral normalization, f(X) is bounded inT[ρ]F = H1

ρ (Γ, g).

Proof. For some family γ1, . . . , γ3N ⊂ Γ r e, the dλγi form a dual basis

of T[ρ]F. Therefore, to bound f(X) we just apply (2.6) to these curves.

3. Main steps in the proof of Theorems 1.4 and 1.7

We continue with the notation of Section 2. As in Section 1.2, let X be thesubset of the arc complex X obtained by removing all faces corresponding


to collections of arcs that do not subdivide the surface S into disks. ByPenner’s work [P] on the decorated Teichmuller space, X is an open ball ofdimension 3|χ|−1, where χ ∈ −N is the Euler characteristic of S. Therefore,in order to prove Theorems 1.4 and 1.7, it is sufficient to prove the followingproposition.

Proposition 3.1. For any convex cocompact ρ ∈ Hom(Γ, G),

(1) the restriction of f to X (resp. of F to CX) takes values in theprojectivized admissible cone adm(ρ) (resp. in Adm+(ρ)),

(2) adm(ρ) is an open ball of dimension 3|χ| − 1 and Adm+(ρ) is homo-topically trivial,

(3) the restrictions f : X → adm(ρ) and F : CX → Adm+(ρ) are proper,(4) the restrictions f : X → adm(ρ) and F : CX → Adm+(ρ) are local

homeomorphims.

Indeed, (3) and (4) imply that the restrictions f : X → adm(ρ) andF : CX → Adm+(ρ) are coverings, and (2) implies that they are trivialcoverings.

3.1. Proof of Proposition 3.1.(1)–(3).

Proof of Proposition 3.1.(1). (See also [PT].) The fact that f(X) ⊂ adm(ρ)means that any infinitesimal strip deformation of ρ, performed on pairwisedisjoint arcs α1, . . . , αk+1 that subdivide S into disks, lengthens every curveat a uniform rate relative to its length. To see that this is true, note that,by compactness, there exist R > 0 and θ ∈ (0, π/2] such that any closedgeodesic γ of S must cross one of the αi at least once every R units oflength, at an angle ≥ θ (because γ cannot exit the convex core). Therefore,if we set ρt := F (tx) ∈ F and denote by γ the element of Γ correspondingto γ, then by (2.3) we have

d

dt

∣∣∣t=0

λγ(ρt)

λγ(ρ)≥M sin θ

R> 0

where the constantM can be taken to be the minimum of the strip widths wαiof (2.2). This proves that any infinitesimal strip deformation f(x) satisfiesthe properness criterion (1.2), and therefore f(X) ⊂ adm(ρ).

Moreover, the bounds θ and R can be taken to hold uniformly when ρand the geodesic representatives αi vary in a compact subset of the de-formation space. Thus, to see that F (CX) ⊂ Adm+(ρ), we can argue byintegration of the infinitesimal relation: for any t0 ≥ 0 and x ∈ X, thevector d

dt

∣∣t=t0

F (tx) ∈ TF (t0x)F belongs to adm(F (t0x)), being for example

realized as an infinitesimal strip deformation along arcs which are bound-ary components of the strips used to produce F (t0x) from ρ. Thereforeλγ(ρt)/λγ(ρ) ≥ eεt for all t ∈ [0, t0], where ε is independent of γ and t. Thisproves F (CX) ⊂ Adm+(ρ).


Proof of Proposition 3.1.(2). By construction (see (1.2)), the positive ad-missible cone of ρ is a convex subset of H1

ρ (Γ, g) and is open, hence itsprojectivization adm(ρ) is an open ball of dimension 3|χ| − 1.

To check that Adm+(ρ) is homotopically trivial, fix k ∈ N and consider amap σ from the sphere Sk to Adm+(ρ). We want to deform σ to a constantmap, among maps valued in Adm+(ρ). Choose a hyperideal triangulation∆ of S. For any t ≥ 0, let σt : Sk → F be the postcomposition of σ by stripdeformations of width t at all edges of ∆ (realized for example by geodesicsexiting the boundary of the convex core of (S, j) perpendicularly). Thenσt(Sk) ⊂ Adm+(ρ) by Proposition 3.1.(1), and for large t the convex core ofany hyperbolic metric corresponding to some σt(q), q ∈ Sk, looks in fact likea collection of near-ideal triangles connected by long, thin isthmi of lengtht + O(1), according to the combinatorics of the dual trivalent graph of ∆.Here the error term O(1) is uniform in q ∈ Sk. The widths of the isthmi

are O(e−t/2) and form coordinates for the convex cocompact Fricke space F.There exists ε > 0, depending on ρ, such that when these coordinates areall ≤ ε, then the metric is in Adm+(ρ). Choose t large enough to make allwidths ≤ ε in the metrics σt(q) (for all points q of Sk), then interpolatelinearly to widths (ε, ..., ε). This proves that σ is homotopically trivial.

In particular, Adm+(ρ) is connected and simply connected; Theorem 1.7will imply that it is actually a ball.

Proof of Proposition 3.1.(3). To establish the properness of the restrictionf : X → adm(ρ), suppose that we have a sequence (xn) which escapes toinfinity in X, but such that f(xn) converges to some class [u] ∈ P(H1

ρ (Γ, g)).We must show that [u] does not lie in adm(ρ). We may work in the unit-peripheral normalization (2.4) of the widthsmγ : indeed, the case of arbitrarywidths reduces to this case up to adapting the sequence (xn) (a changeof widths only alters f projectively inside each simplex). Since f(X) isbounded in H1

ρ (Γ, g) in the unit-peripheral normalization (Proposition 2.6),the sequence (f(xn))n∈N converges to some infinitesimal deformation u inthe projective class [u]. If the supporting cellulation |xn| stabilizes (up topassing to a subsequence) but enough weights wα go to 0 so that the limitingdeformation is in f(X rX), then the corresponding collection of arcs failsto decompose the surface into disks, hence the infinitesimal deformation ufails to lengthen all curves, and [u] does not belong to adm(ρ). Otherwise,the supporting cellulations escape. In this case, let Λ ⊂ S be a Hausdorfflimit of the |xn| (a geodesic lamination, with some isolated leaves escapingin the funnels). For any ε > 0 we can find a simple closed geodesic γ formingangles at most ε/2 with Λ, hence at most ε with |xn| for large n. By (2.6),the corresponding element γ ∈ Γ satisfies

dλγ(u)

λγ(ρ)≤ ε2K.


Since ε is arbitrary, this means u does not satisfy (1.2), so [u] does notbelong to adm(ρ). Thus f is proper.

We next show that the restriction F : CX → Adm+(ρ) is proper. Again,we work in the unit-peripheral normalization (2.4). Suppose a sequence(tnxn) escapes to infinity in CX, where (tn, xn) ∈ R∗+ × X. If tn → +∞,then F (tnxn) escapes to infinity in F, because the boundary length of theconvex core of the corresponding hyperbolic metrics on S is unbounded.We may therefore assume that tn stays bounded, and (up to extracting asubsequence) converges to some t < +∞. If the supports |xn| converge, thenagain the limit in F is of the form [ρ′] = F (tx) with the support |x| disjointfrom some simple closed geodesic γ, hence the corresponding element γ ∈ Γsatisfies λγ(ρ′) = λγ(ρ), which implies [ρ′] /∈ Adm+(ρ). If the supports |xn|diverge, we argue as in the infinitesimal case: for any ε > 0, some simpleclosed geodesic γ forms angles ≤ ε with |xn| for all large n. Proposition 2.5then gives ∑

p∈γ∩|xn|

ϑxn(p) ≤ Kελγ(ρ)

for the corresponding element γ ∈ Γ. Now, consider the representative ofγ in the metric F (tnxn) that agrees with γ outside of the strips and alsoincludes (nongeodesic) segments crossing each strip at constant distancefrom the waist. The length of this representative is exactly

λγ(ρ) + tn∑

p∈γ∩|xn|

ϑxn(p).

Therefore, the length of γ in the metric F (tnxn) is at most (1 +Ktnε)λγ(ρ)and so any limit [ρ′] ∈ F satisfies λγ(ρ′) ≤ (1 + Ktε)λγ(ρ). Since ε isarbitrary, this proves that [ρ′] /∈ Adm+(ρ).

3.2. Reduction of Proposition 3.1.(4). The following claim is a steppingstone to the proof of Proposition 3.1.(4); it will be proved in Section 5. Thenumbering of the statements refers to the codimension of the faces.

Claim 3.2. Let ∆,∆′ be two hyperideal triangulations of S differing by adiagonal switch.

(0) The points f(α), for α ranging over the 3|χ| edges of ∆, are thevertices of a nondegenerate, top-dimensional simplex in P(T[ρ]F), de-noted by f(∆).

(1) The simplices f(∆) and f(∆′) have disjoint interiors in P(T[ρ]F).(2) There exists a choice of geodesic arcs α and waists pα such that

f(∆) ∪ f(∆′) is convex in P(T[ρ]F).

Here, we say that a closed subset of P(T[ρ]F) is convex if it appears convexand compact in some affine chart (note that the whole image f(X) hascompact closure in some affine chart, by Proposition 2.6). We now explainhow Proposition 3.1.(4) follows from Claim 3.2.


Proof of Proposition 3.1.(4) assuming Claim 3.2. Let us prove that f is a lo-cal homeomorphism near any point x ∈ X. The simplicial structure of Xdefines a partition of X into strata, where the stratum of x is the uniquesimplex containing x in its interior. If x belongs to a top-dimensional stra-tum of X, then local homeomorphicity is Claim 3.2.(0). If x belongs to acodimension-1 stratum, then local homeomorphicity is Claim 3.2.(1). Wenow treat the important case that x belongs to a codimension-2 stratum.

Codimension-2 strata of X come in two types. The first type correspondto the decompositions of S into two (hyperideal) squares and 2|χ| − 4 tri-angles. There is nothing to say here: a neighborhood of the image under fof such a stratum of X is subdivided into quadrants covered by the fouradjacent simplices; this is simply a product of two codimension-1 situationsas above.

α1

α2

α3

α4

α5

α1

α2

α3

α4

α5

f(α1)

f(α2)

f(α3)

f(α4)f(α5)

∆ ∆′

f(∆) f(∆′)

f

Figure 2. A pentagon move and its image under f . Theimage is projected along the direction of f(∆0) where ∆0

is a cellulation containing one pentagon and many triangles:thus f(∆0) is represented by the dark point at the center ofthe right panel. In the source (left panel), the arcs αi crossvarious boundary components (dashed) at near-right angles.

The second type of codimension-2 strata of X correspond to the decom-positions of S into one (hyperideal) pentagon and 2|χ|−3 triangles. The linkof such a stratum is a pentagon (the “pentagon moves” between triangula-tions, see Figure 2). Place a Euclidean metric on an affine chart of P(T[ρ]F)containing f(X), so that the simplices f(∆) become endowed with dihedralangles at their codimension-2 faces. By Claim 3.2.(1), we know that the linkat x of the map f is a (piecewise projective) map from a circle to a circle,of degree either 1 or 2, because the image of each of the five segments inthe source circle covers between 0 and π of the target circle: five numbers(θi)i∈Z/5Z in that range (0, π) can add to either 2π or 4π, but to no othermultiple of 2π.


The degree (1 or 2) remains constant as we change the positions of thegeodesic arcs and waists, because one cannot pass continuously from 2π to4π. However, by Claim 3.2.(2) there is at least one choice of geodesic arcsand waists for which the degree is 1. Indeed, by convexity of f(∆) ∪ f(∆′),one pair of consecutive numbers θi, θi+1 ∈ (0, π) has sum ≤ π: the remainingthree numbers cannot add to ≥ 3π, so a total of 4π is impossible. Thereforethe degree is one for these arcs and waists, hence for all arcs and waists.

If x belongs to a stratum of X of codimension d ≥ 3, then we argue byinduction on d. The link map is a map from a (d− 1)-sphere Sd−1 to itself,and it is locally a homeomorphism by induction, hence it is a covering. Butany connected covering of Sd−1 is trivial (a homeomorphism) because Sd−1

is simply connected when d ≥ 3. Therefore f is a local homeomorphism.In the macroscopic case, that the restriction of F to CX is a local home-

omorphism follows by the same argument as for the restriction of f to X.Indeed, as in Proposition 3.1.(1), an infinitesimal perturbation of the weightsof an actual (noninfinitesimal) strip deformation is the same as an infinites-imal deformation performed on the deformed surface.

4. Realizing infinitesimal strip deformations

In Section 3 we explained how Theorems 1.4 and 1.7 reduce to Claim 3.2.Before we prove Claim 3.2 in Section 5, we introduce some notation andformalism that will also be useful in Section 7.

4.1. Killing vector fields on H2. We identify the Minkowski space R2,1

with the Lie algebra g, equipped with its Killing form 〈·|·〉. Note that g alsonaturally identifies with the space of Killing vector fields on H2, i.e. vectorfields whose flow preserves the hyperbolic metric: an element X ∈ g definesthe Killing field

p 7−→ d

dt

∣∣∣t=0

(etX · p) ∈ TpH2.

We shall write X(p) ∈ TpH2 for the vector at p of the Killing field X ∈ g.Recall that X ∈ g is said to be hyperbolic (resp. parabolic, resp. elliptic)if the one-parameter subgroup of G generated by X is hyperbolic (resp.parabolic, resp. elliptic). We view the hyperbolic plane H2 as a hyperboloidin R2,1 and its boundary at infinity ∂∞H2 as the projectivized light cone:

H2 =x ∈ R2,1 | x2

1 + x22 − x2

3 = −1, x3 > 0,(4.1)

∂∞H2 =

[x] ∈ P(R2,1) | x21 + x2

2 − x23 = 0

.

The geodesic lines of H2 are the intersections of the hyperboloid with planesof R2,1. For any p ∈ H2, the tangent space TpH2 naturally identifies with the

linear subspace p⊥ ⊂ R2,1 of vectors orthogonal to p; this linear subspaceis the translate back to the origin of the affine plane tangent at p to thehyperboloid H2 ⊂ R2,1. For any X ∈ g ' R2,1 (seen as a Killing field


on H2),

(4.2) X(p) = X ∧ p ∈ TpH2 = p⊥ ⊂ R2,1 ' g,

where ∧ is the natural Minkowski cross product on R2,1:

(x1, x2, x3) ∧ (y1, y2, y3) := (−x2y3 + x3y2 , −x3y1 + x1y3 , x1y2 − x2y1).

Note that for any x, y ∈ R2,1 the vector x ∧ y ∈ R2,1 is orthogonal to bothx and y, and (x, y, x∧ y) is positively oriented (i.e. satisfies the “right-handrule”). Here is an easy consequence of (4.2).

Lemma 4.1. An element X ∈ R2,1 ' g, seen as a Killing vector field on H2,may be described as follows:

(1) If X is timelike (i.e. 〈X,X〉 < 0), then it is elliptic: it is an infinites-

imal rotation of velocity ±√|〈X,X〉| centered at X√

|〈X,X〉|∈ H2 ⊂ R2,1.

The velocity is positive if X is future-pointing and negative otherwise.(2) If X is lightlike (i.e. 〈X,X〉 = 0 and X 6= 0), then it is parabolic

with fixed point [X] ∈ ∂∞H2 ⊂ P(R2,1).(3) If X is spacelike (i.e. 〈X,X〉 > 0), then it is hyperbolic: it is an in-

finitesimal translation of velocity√〈X,X〉 with axis ` = X⊥ ∩H2 ⊂

R2,1. If v+, v− ∈ X⊥ are future-pointing lightlike vectors repre-senting respectively the attracting and repelling fixed points of X in∂∞H2 ⊂ P(R2,1), then the triple (v+, X, v−) is positively oriented(i.e. satisfies the right-hand rule).

(3′) Let `′ be a geodesic of H2 whose endpoints in ∂∞H2 ⊂ P(R2,1) arerepresented by future-pointing lightlike vectors w1, w2 ∈ R2,1. ThenX is an infinitesimal translation along an axis orthogonal to `′ if andonly if X is spacelike and belongs to span(w1, w2). Endow `′ withthe transverse orientation placing [w1] on the left; then X translatesin the positive (resp. negative) direction if and only if X ∈ R∗+w1 +R∗−w2 (resp. X ∈ R∗−w1 + R∗+w2).

4.2. Bookkeeping for cocycles. We now introduce a formalism for de-scribing cocycles that will be useful in the proofs of Claim 3.2 and Theo-rem 1.6. The basic idea, following Thurston’s description of earthquakes, isto consider deformations of the hyperbolic surface (to be named ϕ) which arelocally isometric everywhere except along some fault lines, where they arediscontinuous. Each such deformation is characterized (up to equivalence)by a map (to be named ψ) describing the relative motion of two pieces ofthe surface adjacent to a fault line.

Consider a geodesic cellulation ∆ of the convex cocompact hyperbolicsurface S = ρ(Γ)\H2, consisting of vertices V , geodesic edges E , and finite-sided polygons T . The elements of E may be geodesic segments, properlyembedded geodesic rays, or properly embedded geodesic lines. A particularcase of interest is when E consists of the geodesic representatives of thesupporting arcs of a strip deformation f(x); in this case all polygons are

hyperideal and V = ∅. Let ∆ be the the preimage of ∆ in the universal


cover H2. The vertices V , edges E , and polygons T of the cellulation ∆are the respective preimages of V ,E , and T . In what follows, we refer to

the elements of T as tiles. We denote by ±E the set of edges of E endowed

with a transverse orientation. For e ∈ ±E , we denote by −e the same edgewith the opposite transverse orientation.

Let us first recall a description of infinitesimal earthquake transforma-tions. For simplicity, we assume that the fault locus is a finite union ofsimple closed geodesics and properly embedded geodesic lines. We maybuild a geodesic cellulation ∆ such that the union of all edges in E contains

the fault locus. Now consider an assignment ϕ : T → g of infinitesimal

motions to the tiles T of the lifted cellulation ∆ (using the interpretation of

Section 4.1). Define a map ψ : ±E → g by assigning to any transversely ori-

ented edge e ∈ ±E the relative motion along that edge: ψ(e) = ϕ(δ′)−ϕ(δ)

where δ, δ′ ∈ T are the tiles adjacent to e, with e transversely oriented fromδ to δ′. The map ϕ defines an infinitesimal left earthquake transformationof S if ψ is ρ-equivariant, i.e.

(4.3) ψ(ρ(γ) · e) = Ad(ρ(γ))ψ(e)

for all γ ∈ Γ and e ∈ ±E , and if for any e ∈ ±E whose projection lies inthe fault locus, ψ(e) is an infinitesimal translation to the left along e (andψ(e) = 0 if e does not project to the fault locus). It is a simple observationthat the infinitesimal deformation of S described by ϕ depends only on therelative motion map ψ; that is, ϕ may be recovered from ψ up to a globalisometry (see below).

We now generalize this description of infinitesimal earthquakes and workwith a larger class of deformations, for which the relative motion betweenadjacent tiles is allowed to be any infinitesimal motion. Consider a ρ-

equivariant map ψ : ±E → g satisfying the following consistency conditions:

• ψ(−e) = −ψ(e) for all e ∈ ±E ;

• the total motion around any vertex is zero: if e1, . . . , ek ∈ ±E arethe transversely oriented edges crossed (in the positive direction) by

a loop encircling a vertex v ∈ V , then∑k

i=1 ψ(ei) = 0.

Under these conditions, ψ defines a cohomology class [u] ∈ H1ρ (Γ, g) as fol-

lows. Choose a base tile δ0 ∈ T and an element v0 ∈ g. Then ψ determines

a map ϕ : T → g by integration: given a tile δ ∈ T , consider a pathp : [0, 1]→ H2 with initial endpoint p(0) in the interior of δ0 and final end-

point p(1) in the interior of δ, such that p(t) avoids the vertices V of ∆; weset

ϕ(δ) := v0 +∑

ψ(e),

where the sum is over all transversely oriented edges e ∈ ±E crossed (inthe positive direction) by the path p(t). This does not depend on the choice


of p, by the second consistency condition above. For any tile δ ∈ T ,

u(γ) := ϕ(ρ(γ) · δ)−Ad(ρ(γ))ϕ(δ)

defines a ρ-cocycle u : Γ → g, independent of δ: we shall say that ϕ is(ρ, u)-equivariant. The cohomology class of u depends only on ψ, not onthe choice of δ0 and v0: indeed, the map integrating ψ with initial data

δ′0 ∈ T and v′0 ∈ g differs from ϕ by the constant vector w0 := v′0 − ϕ(δ′0),and therefore the cocycle it determines differs from u by the coboundaryuw0 : γ 7→ w0 − Ad(ρ(γ))w0. The map ϕ assigns infinitesimal motions to

the tiles in T ; by construction, ψ(e) = ϕ(δ′)−ϕ(δ) for any tiles δ, δ′ adjacent

to an edge e ∈ ±E transversely oriented from δ to δ′.

Let Ψ(±E , g) be the space of ρ-equivariant maps ψ : ±E → g satisfyingthe two consistency conditions above. The integration process ψ 7→ [u] wehave just described defines an R-linear map

(4.4) L : Ψ(± E , g

)−→ H1

ρ (Γ, g).

Note that each tile has trivial stabilizer in Γ, because it is finite-sided and Γis torsion-free. Therefore the map L is onto: any infinitesimal deformation[u] ∈ H1

ρ (Γ, g) of ρ is achieved by some assignment ψ of relative motions.

Indeed, choose a representative in T for each element of T , and choosearbitrary values for ϕ on these representatives. We can extend this to a

(ρ, u)-equivariant map ϕ : T → g, and define ψ(e) := ϕ(δ′) − ϕ(δ) for any

tiles δ, δ′ adjacent to an edge e ∈ ±E transversely oriented from δ to δ′.This map ψ satisfies the consistency conditions above.

4.3. Infinitesimal strip deformations. In our main case of interest, thecellulation ∆ is a geodesic hyperideal triangulation and the edges E of ∆ arethe supporting arcs of an infinitesimal strip deformation. Consider an arc αof S and let α, pα,mα be the chosen geodesic representative, waist, and widthdefining the infinitesimal strip deformation f(α) ∈ H1

ρ (Γ, g). Assuming that

α ∈ E , we define the relative motion map ψα : ±E → g as follows:

• for any transversely oriented lift α ∈ ±E of α, the element ψα(α) ∈ gis the infinitesimal translation of velocity mα along the axis orthog-onal to α at (the lift of) pα, in the positive direction;

• ψα(e) = 0 for any other transversely oriented edge e ∈ ±E .

Recall the map L from (4.4). The following observation is elementary butessential for the proof of Claim 3.2 in Sections 5.1 and 5.2.

Observation 4.2. The relative motion map ψα realizes the infinitesimalstrip deformation f(α), in the sense that L (ψα) = f(α).

In Section 5.2, it will be necessary to work simultaneously with two dif-

ferent hyperideal triangulations ∆ and ∆′. Consider a common refinement

∆′′

of ∆,∆′. For ∆

′,∆′′, we use notation E ′,E ′′ and L ′,L ′′ similar to


Section 4.2. Then there are natural inclusion maps

I : Ψ(±E , g) → Ψ(±E ′′, g) and I ′ : Ψ(±E ′, g) → Ψ(±E ′′, g)

defined as follows: for any ψ ∈ Ψ(±E , g) and e′′ ∈ ±E ′′, set I (ψ)(e′′) :=

ψ(e) if e′′ is contained in an edge e ∈ ±E , with matching transverse ori-entations, or set I (e′′) := 0 otherwise; and similarly for I ′. By usingthese inclusion maps we may compare relative motion maps defined on the

two different triangulations ∆ and ∆′. We consider ψ and ψ′ equivalent if

I (ψ) = I (ψ′). Here are two simple observations:

Observations 4.3. (1) For any ψ ∈ Ψ(±E , g) and ψ′ ∈ Ψ(±E ′, g),

L ′′(I (ψ)) = L (ψ) and L ′′(I ′(ψ′)) = L ′(ψ′).

(2) Let α be an arc of S with geodesic representative α contained in E ∩ E ′.Let ψα and ψ′α be the two relative motion maps realizing f(α) as in Obser-vation 4.2, so that L (ψα) = L ′(ψ′α) = f(α). Then I (ψα) = I ′(ψ′α).

Observation 4.3.(2) means that for any arc α the map ψα is well defined,up to equivalence, independently of the hyperideal triangulation ∆ contain-ing α.

We will also need to compose (i.e. add) infinitesimal strip deformationssupported on arcs that intersect. Suppose that α and α′ are two arcs with

geodesic representatives α and α′ contained in ∆ and ∆′, respectively. We

define the sum ψα + ψα′ to be

ψα + ψα′ := I (ψα) + I ′(ψα′) ∈ Ψ(±E ′′, g).

Clearly L commutes with this operation: L ′′(ψα+ψα′) = L (ψα)+L ′(ψα′).

5. Behavior of f at faces of codimension 0 and 1

We now prove Claim 3.2, using the formalism of Section 4.

5.1. Proof of Claim 3.2.(0). Let ∆ be the geodesic hyperideal triangu-lation whose edges E are the geodesic representatives α, chosen in the defi-nition of the map f , of the arcs α of ∆. We continue with the notation ofSections 4.2 and 4.3. Let us prove that the infinitesimal strip deformationsf(α) ∈ H1

ρ (Γ, g), for α ∈ E , span all of H1ρ (Γ, g). Since dimH1

ρ (Γ, g) = #E ,it is equivalent to show that the f(α) are linearly independent. Supposethat

(5.1)∑α∈E

cα f(α) = 0

for some (cα) ∈ RE , and let us prove that cα = 0 for all α ∈ E . ByObservation 4.2 and linearity of L , the left-hand side of (5.1) is realized by

the ρ-equivariant relative motion of the tiles ψ :=∑

α∈E cαψα : ±E → g,

such that for any transversely oriented lift α ∈ ±E of any α ∈ E ,

(5.2) ψ(α) = cα ψα(α).


Since L (ψ) = 0 by (5.1), there is a map ϕ : T → g integrating ψ thatis (ρ, 0)-equivariant (i.e. ρ-equivariant in the sense of (4.3)). Indeed, as inSection 4.2, choose an arbitrary base tile δ0 and an arbitrary motion of that

tile v0. The map ϕ′ : T → g determined by this initial data is (ρ, uw0)-equivariant for some ρ-coboundary uw0 = (γ 7→ w0 − Ad(ρ(γ))w0). Then

the map ϕ := ϕ′ − w0 : T → g integrates ψ and is (ρ, 0)-equivariant.

Consider an edge α ∈ E , with adjacent tiles δ, δ′ ∈ T . The vectorsv := ϕ(δ) and v′ := ϕ(δ′) encode the infinitesimal motions of the respec-tive tiles δ, δ′. The vector v may be decomposed as v = vt + v`, wherevt ∈ span(α) ⊂ R2,1 is called the transverse motion and v` ⊥ span(α) thelongitudinal motion. By Lemma 4.1.(3), the longitudinal motion v` is an in-finitesimal translation with axis α. Similarly, we decompose v′ as v′ = v′t+v

′`.

By (5.2) and Lemma 4.1.(3′), if we endow α with the transverse orientationfrom δ to δ′, then ψ(α) = v′ − v ∈ span(α), which means that v and v′

impart the same longitudinal motion to α, i.e. v` = v′`. Thus α receives a

well-defined amount√〈v`, v`〉 =

√〈v′`, v′`〉 of longitudinal motion from ϕ,

equal to the longitudinal motion of the tile on either side of the edge; thisamount is invariant under the action of ρ(Γ) because ϕ is (ρ, 0)-equivariant.

It is sufficient to prove that all longitudinal motions of edges of E are zero,because then ϕ = 0 and ψ = 0, and so the linear dependence (5.1) is trivial.Indeed, the three linear forms on R2,1 that vanish on the three boundaryarcs of a tile form a basis of the dual of R2,1 (because the arcs, extendedin P(R2,1), have no common intersection point), hence a Killing field thatimparts zero longitudinal motion to all three arcs must be zero. We nowassume by contradiction that not all longitudinal motions are zero.

Now assume α is an edge receiving maximal longitudinal motion, i.e.that v` = v′` is maximal amongst all edges. Let A,B,C,D,E, F (resp.A,B,C ′, D′, E′, F ′) be the endpoints in ∂∞H2 ⊂ P(R2,1) of all the edges of δ(resp. δ′), cyclically ordered (see Figure 3, left). For convenience, we refer toan edge by its two endpoints, so that e.g. α = AB := H2 ∩ span(A,B). Thefact that the longitudinal motion of AB is at least that of CD means that theimage [v] ∈ P(R2,1) of v lies in a bigon of P(R2,1) bounded by two projectivelines through the point AB ∩ CD, namely the line through AB ∩ CD andAC ∩BD and the line through AB∩CD and AD∩BC. Of the two regionsof P(R2,1) that these lines determine, the correct one is the one containingCD (if [v] is on CD then the longitudinal motion of CD is zero). We referto Figure 3 (left panel), in which the relevant region is shaded. Similarly,the fact that the longitudinal motion of AB is at least that of EF meansthat [v] lies in a region of P(R2,1) bounded by two projective lines throughthe point AB ∩ EF , namely the line through AB ∩ EF and AE ∩BF andthe line through AB∩EF and AF ∩BE. These two conditions determine aquadrilateral Q of P(R2,1) to which [v] must belong. Similarly, [v′] ∈ P(R2,1)must belong to another quadrilateral Q′ of P(R2,1), corresponding to thefact that the longitudinal motion of AB is at least that of C ′D′ or E′F ′.


δ

δ′αA B

C

DE

F

C

DE

F

2c 2d 2e 2f

2/c

2/d

2/e

2/f

C ′D′E′

F ′

Q

Q′

C ′

D′E′

F ′

Q

H

(A)

(B)

(A)

(B)

Projective transformation

Figure 3. Left: view of P(R2,1) and its subset ∂∞H2 in theaffine chart x3 = 1. Right: view in another affine chart,obtained by slicing R2,1 along a plane parallel to span(A,B).

Consider the affine chart of P(R2,1) obtained by slicing R2,1 along theaffine plane parallel to span(A,B) passing through v and v′; note that thisplane contains the origin only if v` = v′` = 0 and we have assumed thisis not the case. The corresponding projective transformation is shown inFigure 3, right panel. In this new affine chart the points A and B are atinfinity, the points C,D,E, F lie in this order on one branch of a hyperbolaH with asymptotes of directions A (horizontal) and B (vertical), and thepoints C ′, D′, E′, F ′ lie in this order on the other branch of H. Consider therestriction of the R2,1 metric to this affine plane. The two asymptotes, whichare lightlike, divide the plane into four quadrants: two of them timelike(namely those containing H) and two of them spacelike. We claim that Qand Q′ lie in opposite, timelike quadrants; this will contradict the fact thatv′ − v = ψ(α) is spacelike. Indeed, by construction the quadrilateral Q isthe intersection of two infinite Euclidean strips: the first strip is the unionof all translates, along the direction (CD), of the rectangle circumscribed to[CD] with sides parallel to the asymptotes; the second strip is the union of alltranslates, along the direction (EF ), of the rectangle circumscribed to [EF ].Without loss of generality, we may assume that C,D,E, F have coordinates(c, 1/c), (d, 1/d), (e, 1/e), (f, 1/f) where 0 < c < d < e < f < +∞. Thenthe four boundary lines of the two Euclidean strips intersect the horizontalaxis at distance 2c < 2d < 2e < 2f from the origin, and the vertical axis atdistance 2/c > 2/d > 2/e > 2/f from the origin. The quadrilateral Q, which


is the intersection of the two strips, therefore lies entirely in the timelikequadrant that contains the branch ofH on which C,D,E, F lie. Similarly, Q′

lies entirely in the opposite quadrant. Therefore the image of ψ(AB) = v′−vin P(R2,1) belongs to the timelike quadrant containing Q′: contradiction.

5.2. Proof of Claim 3.2.(1)–(2). The two hyperideal triangulations ∆and ∆′ have all but one arc in common. Let α (resp. α′) be the arc of∆ (resp. ∆′) that is not an arc of ∆′ (resp. ∆). By Claim 3.2.(0), thesets f(β) : β an arc of ∆ and f(β) : β an arc of ∆′ are both bases ofH1ρ (Γ, g). Therefore there is, up to scale, exactly one linear relation of the

form

(5.3) cα f(α) + cα′ f(α′) +∑

β arc of both∆ and ∆′

cβ f(β) = 0,

where cα, cα′ , cβ ∈ R. Claim 3.2.(1) is equivalent to the inequality cαcα′ > 0.By the nondegeneracy guaranteed by Claim 3.2.(0), it will hold in general ifit holds for one particular choice of geodesic arcs, waists, and widths (usingRemark 2.1). Therefore, it is sufficient to exhibit some choice of geodesicarcs, waists, and widths for which

(5.4)

cα > 0,cα′ > 0,cβ ≤ 0 for all other arcs β of ∆ and ∆′.

The last inequality will imply Claim 3.2.(2).The arcs α, α′ are diagonals of a quadrilateral bounded by four arcs

β1, β2, β3, β4, with α separating β1, β2 from β3, β4 and α′ separating β2, β3

from β4, β1. In the following, we show how to choose geodesic representa-tives, waists, and widths for the arcs of ∆ and ∆′ so that (5.3) becomes

(5.5) f(α) + f(α′)−∑

β∈β1,β2,β3,β4

f(β) = 0.

In particular, all coefficients cβ in (5.3) vanish except for β = βi.

Let β1, . . . , β4 be any geodesic representatives of β1, . . . , β4, respectively,and let R be the hyperideal quadrilateral bounded by these four edges. Let

β1, . . . , β4 be lifts of these edges bounding a lift R of R. The quadrilateral

R is the intersection of H2 with the cone spanned by four spacelike vectorsv1, . . . , v4 ∈ R2,1, where we index the vi so that

βi = H2 ∩ (R+vi + R+vi+1)

for all 1 ≤ i ≤ 4, with indices to be interpreted cyclically modulo 4 through-out the section (i.e. v5 = v1). We now choose the geodesic representatives α

and α′ so that the lifts α and α′ inside R satisfy

α = H2 ∩ (R+v1 + R+v3) and α′ = H2 ∩ (R+v2 + R+v4).

This configuration is achieved, for example, if all the chosen geodesic repre-sentatives of the arcs of S are perpendicular to the boundary of the convex


core: then v1, v2, v3, v4 are dual to the relevant boundary components of thelift of the convex core to H2.

[v1] [v2]

[v3][v4]

δ1

δ2

δ3

δ4

β1

β2

β3

β4

e1

e2

e3e4

[w0]

Figure 4. View of P(R2,1) in the affine chart x3 = 1. The

quadrilateral R is partitioned into four small tiles δ1, δ2, δ3, δ4.The set E ′′ consist of E ∩ E ′ together with the four geodesicrays ei formed by dividing α and α′ in half at their intersec-tion point. A transverse orientation is chosen for each edge.

Let ∆,∆′

be geodesic hyperideal triangulations of S corresponding to∆,∆′, respectively, and containing our chosen geodesic representatives α,α′, βi from above. We now apply the formalism of Section 4.3 to the smallest

geodesic cellulation ∆′′

refining ∆ and ∆′. The vertex set V ′′ of ∆

′′is

just the one point α ∩ α′. The set T ′′ of polygons consists of the 2|χ| − 2hyperideal triangles in T ∩T ′ and of four “small” (nonhyperideal) trianglesarranged around the vertex. The set E ′′ of edges consist of E ∩ E ′ togetherwith the four half-infinite geodesic rays formed by cutting α and α′ in half at

their intersection point. Let V ′′, E ′′, T ′′ be the respective preimages in H2 of

V ′′,E ′′,T ′′. There are four “small” tiles δ1, δ2, δ3, δ4 ∈ T ′′ that partition R:

δi := H2 ∩ (R+vi + R+vi+1 + R+w0),

where w0 = α ∩ α′. For any i, the tile δi is bounded by the infinite edge βitogether with the two half-infinite edges ei and ei+1, where

ei := H2 ∩ (R+vi + R+w0)

(see Figure 4). Note that α = e1 ∪ e3 and α′ = e2 ∪ e4.By multiplying the vi by positive scalars, we may arrange that

(5.6) v1 + v3 = v2 + v4.

Now, defineϕ(δi) := vi+1 − vi


for all i, and extend this to a ρ-equivariant (in the sense of (4.3)) map ϕ :

T ′′ → g, with value 0 outside the ρ(Γ)-orbits of the δi. The corresponding

ρ-equivariant map ψ : ±E ′′ → g describing the relative motion of the tiles,defined as in Section 4.2, satisfies L (ψ) = 0 because ϕ is ρ-equivariant.Therefore, in order to establish (5.5), it is sufficient to see that for someappropriate choice of the strip waists and widths,

(5.7) ψ = ψα + ψα′ −∑

β∈β1,β2,β3,β4

ψβ

in Ψ(±E ′′, g) (see Section 4.3 for the definition of ψα and of the additionoperation).

We first assume that β1, β2, β3, β4 are pairwise distinct. Endow each βiwith the transverse orientation placing δi on the positive side; this makes βiinto an element of ±E ′′. Then

ψ(βi) = ϕ(δi)− 0 = vi+1 − viis, by Lemma 4.1.(3’), an infinitesimal translation along a geodesic of H2

orthogonal to βi at a point pi; the translation direction is negative withrespect to the transverse orientation. We choose the waist pβi ∈ βi to be

the projection to S = ρ(Γ)\H2 of pi and the width mβi =√〈ψ(βi), ψ(βi)〉

to be the velocity of the infinitesimal translation ψ(βi). With these choices,Observation 4.2 implies that

ψ(βi) = −ψβi(βi),where ψβi is the relative motion map for the strip deformation along βi (seeSection 4.3). We transversely orient the ray ei from δi−1 to δi (see Figure 4).By (5.6),

ψ(ei) = ϕ(δi)− ϕ(δi−1) = vi+1 − 2vi + vi−1 = vi+2 − vi.By Lemma 4.1.(3’), this implies that ψ(ei) is an infinitesimal translationalong a geodesic of H2 orthogonal to ei at some point qi; the direction oftranslation is positive with respect to the transverse orientation. Note thatψ(ei) = ψ(−ei+2), hence qi = qi+2. We choose the waist pα ∈ α to be theprojection to S of q1 = q3, and the width mα > 0 to be the velocity of theinfinitesimal translation ψ(±e1) = ψ(∓e3). Similarly, we choose the waistand width for α′ to be defined by ψ(±e2) = ψ(∓e4). Then, we have

ψ(e1) = ψα(e1), ψ(e2) = ψα′(e2),

ψ(e3) = ψα(e3), ψ(e4) = ψα′(e4),

where we interpret ψα, ψα′ as elements of Ψ(E ′′, g) as in Section 4.3. Since

ψ,ψα, ψα′ , ψβi all take value 0 outside the ρ(Γ)-orbits of the βi and ei,we conclude that (5.7) holds. This establishes (5.5), hence (5.4), henceClaim 3.2.(1)–(2), in the case that β1, β2, β3, β4 are pairwise distinct.


In the case that some of the βi are equal, we still define ϕ as above. For1 ≤ i ≤ 4, if βi is not equal to any other βj , then we choose the waist pβi and

the width mβi as above. If βi = βj for some 1 ≤ j ≤ 4, then ρ(γ) · βi = −βjfor some γ ∈ Γ, and

ψ(−βj) = ϕ(ρ(γ) · δi)− ϕ(δj) = Ad(ρ(γ))ϕ(δi)− ϕ(δj)

is the sum of two infinitesimal translations orthogonal to βj , both pos-

itive with respect to the transverse orientation of βj . Therefore, using

Lemma 4.1.(3’), we see that ψ(−βj) is again a positive infinitesimal transla-

tion orthogonal to βj . We choose ψβi = ψβj to have waist and width defined

by ψ(−βj), so that ψβj (βj) = −ψ(βj). Then (5.7) holds as above. Thiscompletes the proof of Claim 3.2.(1)–(2).

Proposition 3.1 is proved, as well as Theorems 1.4 and 1.7.

6. Examples

(a) (b) (c) (d)

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

3

3

3

3

3

3

Figure 5. Four surfaces of small complexity (top) and theirarc complexes, mapped under f to the closure of adm(ρ) inan affine chart of PH1

ρ (Γ, g) (bottom). Some arcs are labeledby Arab numerals.

Four surfaces (two of them orientable) have a 2-dimensional arc complex.They are represented in Figure 5. Here we summarize some elementaryfacts about their arc complexes X, and how X relates to the geometry ofadm(ρ) for ρ the holonomy representation of a convex cocompact hyperbolicstructure on the surface. Margulis spacetimes whose associated hyperbolicsurface is of one of these four topological types were studied by Charette–Drumm–Goldman in [CDG1, CDG3]; they give a similar tiling of adm(ρ)according to which isotopy classes of crooked planes embed disjointly in theMargulis spacetime.

(a) Thrice-holed sphere: The arc complex X has 6 vertices, 9 edges, 4faces. Its image under f contains the closure of adm(ρ), a triangle whosesides stand in natural bijection with the three boundary components of the


surface S = ρ(Γ)\H2: an infinitesimal deformation u of ρ lies in a side ofthe triangle if and only if it does not alter the length of the correspondingboundary component, to first order.

(b) Twice-holed projective plane: The arc complex X has 8 vertices,13 edges, 6 faces. Its image f(X) contains the closure of f(X) = adm(ρ)(a quadrilateral). The horizontal sides of the quadrilateral correspond toinfinitesimal deformations u that fix the length of a boundary component.The vertical sides correspond to infinitesimal deformations which fix thelength of one of the two simple closed curves running through the half-twist.

(c) Once-holed Klein bottle: The arc complex X is infinite, with onevertex of infinite degree and all other vertices of degree either 2 or 5. Theclosure of f(X) is an infinite-sided polygon with sides indexed in Z ∪ ∞.The exceptional side has only one point in f(X), and corresponds to infini-tesimal deformations that fix the length of the only nonperipheral, two-sidedsimple closed curve γ, which goes through the two half-twists. The group Znaturally acts on the arc complex X, via Dehn twist along γ. All nonexcep-tional sides are contained in f(X) and correspond to deformations which fixthe length of some curve, all these curves being related by some power ofthe Dehn twist along γ.

(d) Once-holed torus: The arc complex X is infinite, with all vertices ofinfinite degree (it is known as the Farey triangulation). Arcs are parameter-ized by P1(Q). The closure of f(X) contains infinitely many segments in itsboundary. These segments, also indexed by P1(Q), are in natural correpon-dence with the simple closed curves. Only one point of each side belongsto f(X): namely, the strip deformation along a single arc, which lengthensall curves except the one curve disjoint from that arc. The group GL2(Z)acts on X, transitively on the vertices, via the mapping class group of theonce-holed torus. See [GLMM] for more about the closure of the admissiblecone in this case.

Note that in cases (c) and (d), no edge of adm(ρ) corresponds to theboundary of the surface, because there is only one boundary component: anynonzero deformation of ρ which weakly lengthens all closed curves, strictlylengthens the boundary (see Proposition 2.6).

7. Fundamental domains in Minkowski 3-space

In this section, we deduce Theorem 1.6 (the Crooked Plane Conjecture,assuming convex cocompact linear holonomy) from Theorem 1.4 (the pa-rameterization by the arc complex of Margulis spacetimes with fixed con-vex cocompact linear holonomy). To begin, we review the construction ofcrooked planes in Minkowski space, originally due to Drumm [D].

7.1. Crooked planes in R2,1. A crooked plane in R2,1, as defined in [D],is the union of


• a stem, which is the union of all causal (i.e. timelike or lightlike) linesof a given timelike plane that pass through a given point, called thecenter ;• two wings, which are two disjoint open lightlike half-planes whose re-

spective boundaries are the two (lightlike) boundary lines of the stem.

Let us fix some notation. We see H2 as a hyperboloid in R2,1 as in (4.1).For any future-pointing lightlike vector v0 ∈ R2,1, we denote by W(v0) theleft wing associated with v0: by definition, this is the connected componentof v⊥0 rRv0 consisting of (spacelike) vectors w that lie “to the left of v0 seenfrom H2”, i.e. such that (v0, w, v) is positively oriented for any v ∈ H2 ⊂ R2,1.For any geodesic line ` of H2, with endpoints in ∂∞H2 ⊂ P(R2,1) representedby future-pointing lightlike vectors v+, v− ∈ R2,1, we denote by C(`) the leftcrooked plane centered at the origin associated with `: by definition, this isthe union of the stem

S(`) := w ∈ span(`) ⊂ R2,1 | 〈w |w〉 ≤ 0and of the wings W(v+) and W(v−) (see Figure 6). A general left crookedplane is just a translate C(`) + v of such a set C(`) by some vector v ∈ R2,1.The images of left crooked planes under the orientation-reversing linear mapx 7→ −x are called right crooked planes; we will not work directly with themhere.

S(`)

W(v+)W(v−)

v+

v−

Figure 6. A left crooked plane in R2,1

Thinking of R2,1 ' g as the set of Killing vector fields on H2 as in Sec-tion 4.1 and using Lemma 4.1, we can describe C(`) as follows:

• the interior of the stem S(`) is the set of elliptic Killing fields on H2

whose fixed point belongs to `;• the lightlike line Rv+, in the boundary of the stem S(`), is the set of

parabolic Killing fields with fixed point [v+] ∈ ∂∞H2, and similarlyfor v−;• the wingW(v+) is the set of hyperbolic Killing fields with attracting

fixed point [v+] ∈ ∂∞H2, and similarly for v−.


In other words, C(`) is the set of Killing fields on H2 with a nonrepellingfixed point in `, where ` is the closure of ` in H2 ∪ ∂∞H2.

Any crooked plane divides R2,1 into two connected components. Given atransverse orientation of `, the positive crooked half-space H+(`) (resp. thenegative crooked half-space H−(`)) is the connected component of R2,1rC(`)consisting of nontrivial Killing fields on H2 with a nonrepelling fixed pointin (H2 ∪ ∂∞H2) r ` lying on the positive (resp. negative) side of `.

7.2. Disjointness of crooked half-spaces in R2,1. In order to build fun-damental domains in R2,1 for proper actions of free groups, it is importantto understand when two crooked planes are disjoint. A complete disjoint-ness criterion for crooked planes was given by Drumm–Goldman in [DG2].More recently, the geometry of crooked planes and crooked half-spaces wasstudied in [BCDG]. We now recall a sufficient condition due to Drumm.

Let ` be a transversely oriented geodesic of H2 and let v+, v− ∈ R2,1 befuture-pointing lightlike vectors representing the endpoints of ` in ∂∞H2,with [v+] lying to the left for the transverse orientation. We shall use thefollowing terminology.

Definition 7.1. The open cone R∗+v+−R∗+v− of span(v−, v+) = span(`) iscalled the stem quadrant of the transversely oriented geodesic `, denoted bySQ(`).

By Lemma 4.1.(3’), the stem quadrant SQ(`) consists of all infinitesimaltranslations of H2 whose axis is orthogonal to ` and oriented in the posi-tive direction. The following sufficient condition for disjointness of crookedplanes was observed by Drumm:

Proposition 7.2 (Drumm [D]). Let `, `′ be two disjoint geodesics of H2,transversely oriented away from each other, and let v, v′ ∈ R2,1.

• If v ∈ SQ(`) and v′ ∈ SQ(`′), then H+(`) + v ⊂ H−(`′) + v′.

• If in addition v ∈ SQ(`) or v′ ∈ SQ(`′), then H+(`)+v ⊂ H−(`′)+v′;in particular, the crooked planes C(`) + v and C(`′) + v′ are disjoint.

Proof. It is clear from the definitions in terms of nonrepelling fixed pointsof Killing fields that H+(`) ⊂ H−(`′). For v ∈ SQ(`), one checks that

H+(`)+v ⊂ H+(`). Similarly, for v′ ∈ SQ(`′) we have H+(`′)+v′ ⊂ H+(`′),or equivalently H−(`′) + v′ ⊃ H−(`′), and the first assertion follows. Thesecond assertion follows after observing that C(`) ∩ C(`′) = 0.

Conversely, for w ∈ R2,1, we have C(`) ∩ (C(`′) + w) = ∅ if and only if

w ∈ SQ(`′) − SQ(`) or w ∈ SQ(`′) − SQ(`) [DG2, BCDG]. Thus the spaceof directions in one can translate C(`′) to make it disjoint from C(`) is anopen cone of R2,1 with a quadrilateral basis.

7.3. Drumm’s strategy. In the early 1990s, Drumm [D] introduced astrategy to produce proper affine deformations u of ρ. We now briefly recallit; see [ChaG] for more details.


Begin with a convex cocompact representation ρ ∈ Hom(Γ, G). Then ρ(Γ)is a Schottky group, playing ping pong on H2: there is a fundamental domainF in H2 for the action of ρ(Γ) that is bounded by finitely many geodesics`1, `

′1, . . . , `r, `

′r with `′i = ρ(γi) · ì for some γi ∈ Γ. The elements γ1, . . . , γr

generate the free group Γ. The corresponding left crooked planes centeredat the origin in R2,1 satisfy C(`′i) = ρ(γi) · C(ì). Now orient transverselyeach geodesic ì or `′i away from F and translate the corresponding crookedplane C(ì) or C(`′i) by a vector vi or v′i in its stem quadrant. By Proposi-tion 7.2, the resulting collection of crooked planes is pairwise disjoint andbounds a closed region R in R2,1. The Minkowski isometries that identifyopposite pairs of crooked planes generate an affine deformation Γρ,u of ρ(Γ),where u(γi) = v′i − ρ(γi) · vi for all 1 ≤ i ≤ r. (In other words, u comesfrom propagating the movement of the original crooked planes equivariantlyby translating, not only each crooked plane, but actually the whole closedpositive crooked half-space it bounds.) By construction, R is a fundamentaldomain for the action of Γρ,u on the union Γρ,u · R of all translates of R.Drumm proved the following.

Theorem 7.3 (Drumm [D]). In the setting above, Γρ,u · R = R2,1.

In particular, Γρ,u acts properly on R2,1 and R is a fundamental polyhe-dron for this action.

In summary, given a fundamental domain in H2 and appropriate motionsof the corresponding crooked planes, Drumm’s procedure yields a propercocycle u. The idea of the proof of Theorem 1.6 is that Theorem 1.4, cor-rectly interpreted, gives an inverse to Drumm’s procedure: an infinitesimalstrip deformation determines disjoint geodesics in H2 (namely the preimageof the support of the strips) and relative motions of the corresponding leftcrooked planes in R2,1. A fundamental domain in R2,1 bounded by crookedplanes is found by choosing a fundamental domain in H2 bounded by someof the arcs supporting the strip deformation.

7.4. Proof of Theorem 1.6: the Crooked Plane Conjecture. By[FG], any discrete subgroup of SO(2, 1) n R3 acting properly discontinu-ously on R2,1 is of the form Γρ,u as in (1.1). We assume that ρ is convexcocompact. By [GLM1] or [DGK1], the cocycle u belongs to the admissiblecone of ρ (Definition 1.1). Note that replacing u with −u amounts to conju-gating Γρ,u by the orientation-reversing linear map x 7→ −x, which maps leftcrooked planes to right crooked planes. Therefore, it is sufficient to considerthe case that u belongs to the positive admissible cone of ρ, and to provethat in this case there exists a fundamental domain in R2,1 for Γρ,u that isbounded by finitely many left crooked planes.

By Theorem 1.4, the cocycle u is an infinitesimal strip deformation sup-ported along some collection E of geodesic arcs α on S = ρ(Γ)\H2, which cutthe surface into disks. We use the notation and formalism of Section 4. By


Observation 4.2, the infinitesimal strip deformation u is described by a (ρ, u)-

equivariant assignment ϕ : T → g of infinitesimal motions to the tiles, such

that for any tiles δ, δ′ adjacent to an edge α ∈ ±E transversely oriented fromδ′ to δ, the Killing vector field ψ(α) = ϕ(δ)−ϕ(δ′) is an infinitesimal trans-lation of H2 orthogonal to α, in the positive direction; in other words, ψ(α)belongs to the stem quadrant SQ(α) (Definition 7.1) by Lemma 4.1.(3’). To

any α ∈ E we associate the crooked plane Dα := C(α) + vα where

(7.1) vα :=ϕ(δ) + ϕ(δ′)

2.

One could think of vα as the motion of the edge α under the infinitesimaldeformation u, which we take to be the average of the motions of the adjacenttiles. Since ϕ is (ρ, u)-equivariant, the map α 7→ Dα is (ρ, (ρ, u))-equivariant,

meaning that Dρ(γ)·α = ρ(γ) · Dα + u(γ) for all α ∈ E and γ ∈ Γ.

We claim that the crooked planes Dα, α ∈ E , are pairwise disjoint. In-deed, consider two adjacent edges α, α′ bounding tiles δ, δ′, δ′′ as in Figure 7.Transversely orient α from δ′ to δ, and α′ from δ′′ to δ′, so that the positivehalf-plane of α in H2 (i.e. the connected component of H2 r α lying on thepositive side of α for the transverse orientation) is contained in that of α′.Note that

vα − ϕ(δ′) =1

2ψ(α) and vα′ − ϕ(δ′) =

1

2ψ(−α′).

Since ψ(−α′) ∈ SQ(−α′) and ψ(α) ∈ SQ(α), Proposition 7.2 implies

H+(α) + vα − ϕ(δ′) ⊂ H−(−α′) + vα′ − ϕ(δ′),

hence

(7.2) H+(α) + vα ⊂ H−(−α′) + vα′ = H+(α′) + vα′ .

In particular, the crooked planes Dα and Dα′ are disjoint whenever α, α′

border the same tile. Now, let α, α′ ∈ E be any distinct edges. Assign

δ′

αα′

δδ′′

Figure 7. Two adjacent edges α, α and tiles δ, δ′, δ′′ as inthe proofs of Theorems 1.6 and 1.8

transverse orientations so that the positive half-plane of α in H2 is containedin that of α′. Then there is a sequence of transversely oriented edges α =


e0, e1, . . . , eN = α′ ∈ ±E such that any consecutive edges ei, ei+1 bordera common tile and the positive half-plane of ei in H2 is contained in thatof ei+1. By induction and (7.2),

(7.3) H+(α) + vα ⊂ H+(α′) + vα′ .

In particular, the crooked planes Dα and Dα′ are disjoint.To conclude, we note that since the arcs supporting the infinitesimal strip

deformation u cut the surface S = ρ(Γ)\H2 into disks, we may choose arcs

α1, α′1, . . . , αr, α

′r from E that bound a fundamental domain F in H2 for the

action of ρ(Γ), with α′i = ρ(γi) · αi for some γi ∈ Γ. The elements γ1, . . . , γrgenerate the free group Γ. By (7.3), the crooked planes Dαi and Dα′

i, for

1 ≤ i ≤ r, are pairwise disjoint and bound a closed, connected region Rin R2,1. For any i, the element (ρ(γi), u(γi)) ∈ Γρ,u identifies Dαi with Dα′

i.

Therefore, R is a fundamental domain for the action of Γρ,u on Γρ,u · R.That Γρ,u · R = R2,1 then follows by Theorem 7.3, or alternatively from the

following lemma, which implies that the crooked planes Dα, α ∈ E , do notaccumulate on any set.

Lemma 7.4. For any p ∈ H2 and any sequence (αn) ∈ E N going to infinity,

inf‖Xn(p)‖ | Xn ∈ Dαn

−→

n→+∞+∞.

Proof. It is enough to treat the case that there exists a sequence (δn)n∈N of

distinct elements of T (tiles) such that p ∈ δ0 and αn ∈ E is adjacent toδn and δn−1 for all n ≥ 1. We orient αn transversely towards δn. Considera Killing field Xn = Yn + vαn ∈ Dαn , where Yn ∈ C(αn). There exists anonrepelling fixed point qn of Yn in the closure of αn, so the component of Ynalong the oriented line pqn (oriented towards qn) is nonnegative. Moreover,for each i ≤ n the line pqn crosses αi in the positive direction (at a pointri), so the Killing field ψ(αi) has nonnegative component along pqn, equal to‖ψ(αi)(ri)‖ sin]ri(pqn, αi). In fact, for 2 ≤ i ≤ n− 1 the angle ]ri(pqn, αi)is bounded below by a positive constant depending only on the geometryof the tiles, and ‖ψ(αi)(ri)‖ is at least the width of the strip deformationat αi. It follows that the component along pqn of the Killing field

Xn = Yn + ϕ(δ0) + ψ(α1) + ψ(α2) + · · ·+ ψ(αn−1) +1

2ψ(αn)

goes to infinity as n→ +∞. This completes the proof.

Remark 7.5. In (7.1) above, we could have taken

vα := (1− t)ϕ(δ) + t ϕ(δ′)

for an arbitrary fixed t ∈ (0, 1), not necessarily t = 1/2; the proof wouldhave worked the same way. For each arc supporting the strip deformation,there is an interval’s worth of parallel crooked planes (for t varying in (0, 1)),each embedded in the Margulis spacetime. No two of these crooked planesintersect, except that two crooked planes corresponding to the same edgeare tangent along subsets of their stems.


8. Fundamental domains in anti-de Sitter 3-space

In this section we introduce piecewise totally geodesic surfaces in AdS3

analogous to crooked planes. We then establish a sufficient condition fordisjointness similar to Proposition 7.2, and prove Theorem 1.8.

8.1. AdS crooked planes. As in Minkowski space, we define a crookedplane in AdS3 to be the union of three pieces:

• a stem, defined to be the union of all causal (i.e. timelike or lightlike)geodesics of a given timelike plane of AdS3 that pass through a givenpoint, called the center of the AdS crooked plane;• two wings, defined to be two disjoint open lightlike half-planes of

AdS3 whose respective boundaries are the two (lightlike) boundarylines of the stem.

Figure 8. A left AdS crooked plane, seen in two differentaffine charts of P3(R) ⊃ AdS3. The stem is a bigon whose clo-sure meets the boundary of AdS3 in two points. On the left,these two points are at infinity; on the right, the center of thestem is at infinity. Each wing is itself a bigon, bounded by aline contained in the boundary of AdS3 and a line of the stem.

As in Minkowski space, an AdS crooked plane centered at the identity isdetermined by a geodesic line ` of H2 and a choice of orientation (left orright): we denote by C(`) the left AdS crooked plane centered at e ∈ G0

associated with `, which is described explicitly as follows (see Figure 8):

• the interior of the stem S(`) of C(`) is the set of elliptic elementsh ∈ G0 whose fixed point belongs to `;• the boundary of the stem S(`) is the set of parabolic elements h ∈ G0

fixing one of the two endpoints [v+], [v−] of ` in ∂∞H2;• the wings of C(`) are W(v+) and W(v−), where W(v+) is the set of

hyperbolic elements h ∈ G0 with attracting fixed point [v+], and simi-larly for v−.

In other words, C(`) is the set of orientation-preserving isometries of H2 (i.e.elements of G0 ' AdS3) with a nonrepelling fixed point in `, where ` is againthe closure of ` in H2 ∪ ∂∞H2, as in Section 7.1. Note that this is exactly


the image under the exponential map of the crooked plane C(`) ⊂ R2,1 fromSection 7.1. We refer to Goldman [G] for an explanation of the relationshipbetween Minkowski crooked planes and AdS crooked planes in the contextof Einstein geometry.

A general left AdS crooked plane is a G0-translate of C(`) for some geo-desic ` of H2. The image of left AdS crooked planes under the orientation-reversing isometry g 7→ g−1 of AdS3 are called right AdS crooked planes.

Similarly to the Minkowski setting, an AdS crooked plane divides AdS3

into two connected components (see Figure 8). Note that by contrast atimelike geodesic plane in AdS3 does not divide AdS3 into two components:it is one-sided (topologically a Mobius strip). Given a transverse orientationof `, we denote by H+(`) (resp. H−(`)) the connected component of AdS3 rC(`) consisting of nontrivial elements g ∈ G0 with a nonrepelling fixed pointin (H2 ∪ ∂∞H2) r ` lying on the positive (resp. negative) side of `.

8.2. Disjointness for left AdS crooked planes. Here is a sufficient con-dition for disjointness analogous to Proposition 7.2 (see Figure 9); as before,SQ(`) denotes the stem quadrant of ` (Definition 7.1).

Proposition 8.1. Let `, `′ be two disjoint geodesics of H2, transversely ori-ented away from each other. For any g ∈ exp(SQ(`)) and g′ ∈ exp(SQ(`′)),

gH+(`) ⊂ g′H−(`′)

and the crooked planes g C(`) and g′ C(`′) are disjoint.

Figure 9. Two disjoint AdS crooked planes, green and red,in an affine chart of P3(R) truncated above and below. Thesix singularities are marked by paler dots (hollow if hidden),the centers of the crooked planes being square dots. Dual toeach stem vertex is a copy of H2, part of whose boundary atinfinity of AdS3 is also shown (blue ellipse arcs).

Proposition 8.1 is an immediate consequence of the following.


Lemma 8.2. Let ` be any transversely oriented geodesic of H2. Then forany g ∈ exp(SQ(`)), we have gH+(`) ⊂ H+(`).

Lemma 8.2 means that for any hyperbolic g ∈ G0 with translation axisorthogonal to ` in H2, oriented in the positive direction with respect to thetransverse orientation of `, and for any h ∈ G0 with a nonrepelling fixedpoint in the closed positive half-plane of `, the element gh ∈ G0 has anonrepelling fixed point in the open positive half-plane of `.

Proof of Lemma 8.2. Let h ∈ G0 have a nonrepelling fixed point in theclosed positive half-plane L+ of `, and h′ have a nonrepelling fixed point inthe negative half-plane L−. The lemma says that if g ∈ G0 has translationaxis orthogonal to `, oriented towards L+, then gh 6= h′.

Note that g moves any point x of L+ away from any point x′ of L− by at

least δ := 2 log cosh λ(g)2 > 0 units of length, where λ(g) is the translation

length of g. This is also true for x, x′ horoballs centered in ∂∞L+ and ∂∞L

−,since the distance function to a horoball is a limit of distance functions topoints (up to constants). Thus, if x is a singleton or horoball stable underh, and x′ a singleton or horoball stable under h′, then

d(gh · x, h′ · x′) ≥ d(x, x′) + δ.

It follows that gh cannot equal h′.

Remark 8.3. In Minkowski space, a crooked plane C(`) meets any translateof itself; in particular, for v ∈ SQ(`), the intersection C(`)∩ (C(`) + v) is theunion of two affine subcones of the stem. On the other hand, Lemma 8.2shows that in AdS3, the crooked plane g C(`) for g ∈ exp(SQ(`)) is disjointfrom C(`); it is obtained from C(`) by sliding and tilting (see Figure 9).

Remark 8.4. The Minkowski space R2,1 and the anti-de Sitter space AdS3

can both be embedded into the 3-dimensional Einstein space Ein3. Theclosure in Ein3 of a Minkowski or AdS crooked plane is a crooked surfacein the sense of [F]: see [G]. Note that Drumm’s strategy from Section 7.3has recently been carried out in the Einstein setting in [CFL], although nocomplete disjointness criterion is known for the moment.

8.3. Crooked fundamental domains in AdS obtained from strip de-formations. We now deduce Theorem 1.8 from Theorem 1.7.

By [Ka], up to switching j and ρ we may assume that j is “uniformlylonger” than ρ in the sense that (1.3) is satisfied. Note that switching jand ρ amounts to conjugating Γρ,j by the orientation-reversing isometryg 7→ g−1, which maps left AdS crooked planes to right AdS crooked planes.Assuming that j is “uniformly longer” than ρ, we shall prove that thereexists a fundamental domain in AdS3 for Γρ,j that is bounded by finitelymany left AdS crooked planes.

By Theorem 1.7, we can realize j as a strip deformation of ρ supportedalong some collection E of geodesic arcs α on S = ρ(Γ)\H2, which cut thesurface into disks. We proceed as in the infinitesimal case, again using the


notation of Section 4. Similarly to Observation 4.2, the strip deformation

taking ρ to j is described by an assignment Φ : T → G0 of motions to thetiles which satisfies the following:

• Φ is (ρ, (ρ, j))-equivariant: for all γ ∈ Γ and δ ∈ T ,

Φ(ρ(γ) · δ) = j(γ) Φ(δ) ρ(γ)−1;

• given a transversely oriented arc α ∈ ±E bordering tiles δ, δ′ on thepositive and negative side respectively, the relative displacement

Ψ(α) := Φ(δ′)−1Φ(δ) ∈ G0

is hyperbolic with translation axis orthogonal to α, oriented in thepositive direction, that is Ψ(α) ∈ exp(SQ(α)) by Lemma 4.1.(3’).

To any α ∈ E we associate the AdS crooked plane Dα := gα C(α) where

(8.1) gα := Φ(δ)√

Φ(δ)−1 Φ(δ′) = Φ(δ′)√

Φ(δ′)−1 Φ(δ).

Here the square root of a hyperbolic element denotes the hyperbolic elementwith the same oriented axis but half the translation length. One could thinkof gα as the motion of the edge α under the strip deformation, which wetake to be the average of the motions of the adjacent tiles. Since Φ is(ρ, (ρ, j))-equivariant, Ψ is (ρ, (ρ, ρ))-equivariant and

Dρ(γ)·α = j(γ)Dα ρ(γ)−1

for all γ ∈ Γ and α ∈ E . We claim that the Dα, for α ∈ E , are pairwisedisjoint. Indeed, consider two adjacent edges α, α′ bounding tiles δ, δ′, δ′′ asin Figure 7. Transversely orient α from δ′ to δ, and α′ from δ′′ to δ′ so thatthe positive half-plane of α in H2 is contained in that of α′. Then

Φ(δ′)−1gα =√

Ψ(α), Φ(δ′)−1gα′ =√

Ψ(α′)−1 =√

Ψ(−α′).Since Ψ(α) ∈ exp(SQ(α)) and Ψ(−α′) ∈ exp(SQ(−α′)), Proposition 8.1 im-plies

Φ(δ′)−1gαH+(α) ⊂ Φ(δ′)−1gα′H−(−α′),hence

(8.2) gαH+(α) ⊂ gα′H−(−α′) = gα′H+(α′).

This shows in particular that the crooked planes Dα and Dα′ are disjointwhenever α, α′ border the same tile. As in the Minkowski setting, induction

and (8.2) allow us to conclude that for any edges α, α′ ∈ ±E , transverselyoriented so that the positive half-plane of α in H2 is contained in that of α′,

(8.3) H+(α) ⊂ H+(α′).

A candidate fundamental domain R is then defined as the intersection of thecrooked half-spaces corresponding to the half-planes defining a fundamentaldomain in H2 for ρ(Γ). The proof concludes by showing that Γρ,j ·R = AdS3.This is implied by the following analogue of Lemma 7.4, which shows that

the crooked planes Dα, α ∈ E , do not accumulate on any set.


Lemma 8.5. For any p ∈ H2 and any sequence (αn) ∈ E N going to infinity,

infd(p, gn · p) | gn ∈ Dαn

−→

n→+∞+∞.

Proof. Let α, α′ ∈ E be distinct edges bounding a common tile δ. Then forany x ∈ α, y ∈ α′, a simple calculation shows that d(gαx, gα′y) ≥ d(x, y) +

(δ(α) + δ(α′))/2, where δ(α) := 2 log cosh λ(Ψ(α))2 is a function of the width

of the strip at α′ (as in the proof of Lemma 8.2). The inequality also holds ifx and/or y is a horoball centered at an endpoint of α and/or α′ respectively.

Let α, α′ ∈ E be two edges separated by N tiles and let δ0 be the minimumof the finitely many values δ(α). Then by induction on N , we have thatd(gαx, gα′y) ≥ d(x, y) +Nδ0 holds for x any point of α or horoball centeredon an endpoint of α and y any point of α′ or horoball centered on an endpointof α′. We now fix the edge α and a point p ∈ α, and allow the edge α′ to vary.Without loss in generality we may assume gα = 1. Let h′ = gα′h ∈ Dα′ ,where h ∈ C(α′). Then h either fixes a point y ∈ α′ or preserves a horobally centered at an endpoint of α′. In either case, by the above, we have thatd(p, gα′hy) ≥ d(p, gα′y) ≥ d(p, y)+Nδ0. It follows by the triangle inequalitythat d(p, h′p) ≥ Nδ0. The result follows.

Remark 8.6. In (8.1), we could have taken

gα := Φ(δ)(Φ(δ)−1 Φ(δ′)

)t= Φ(δ′)

(Φ(δ′)−1 Φ(δ)

)1−t ∈ G0

for an arbitrary fixed t ∈ (0, 1), not necessarily t = 1/2; the proof wouldhave worked the same way (where gt = exp(t log g).)

References

[Ab] H. Abels, Properly discontinuous groups of affine transformations : a survey,Geom. Dedicata 87 (2001), p. 309–333.

[Au] L. Auslander, The structure of compact locally affine manifolds, Topology 3(1964), p. 131–139.

[BM] J. Brock, H. Masur, Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity, Geom. Topol. 12 (2008), p. 2453–2495.

[BCDG] J.-P. Burelle, V. Charette, T. Drumm, W. M. Goldman, Crooked half-spaces, Enseign. Math. (2), to appear.

[CDG1] V. Charette, T. Drumm, W. M. Goldman, Affine deformations of the three-holed sphere, Geom. Topol. 14 (2010), p. 1355–1382.

[CDG2] V. Charette, T. Drumm, W. M. Goldman, Finite-sided deformation spacesof complete affine 3-manifolds, preprint, arXiv:1107.2862.

[CDG3] V. Charette, T. Drumm, W. M. Goldman, Proper affine deformations oftwo-generator Fuchsian groups, in preparation.

[ChaG] V. Charette, W. M. Goldman, Affine Schottky groups and crooked tilings,Contemp. Math., Amer. Math. Soc. 262 (2000), p. 69–98.

[CFL] V. Charette, D. Francoeur, R. Lareau-Dussault, Fundamental polyhedrain the Einstein Universe, preprint, arXiv:1307:6531.


[ChoG] S. Choi, W. M. Goldman, Topological tameness of Margulis spacetimes,preprint, 2013.

[DGK1] J. Danciger, F. Gueritaud, F. Kassel, Geometry and topology of completeLorentz spacetimes of constant curvature, preprint, arXiv:1306.2240.

[DGK2] J. Danciger, F. Gueritaud, F. Kassel, Geometry, topology, and modulispaces of general Margulis spacetimes, in preparation.

[DT] B. Deroin, N. Tholozan, Dominating surface group representations by Fuch-sian ones, preprint, arXiv:1311.2919.

[D] T. Drumm, Fundamental polyhedra for Margulis space-times, Topology 21(1992), p. 677–683.

[DG1] T. Drumm, W. M. Goldman, Crooked Planes, Electronic Research Announce-ments of the American Math. Soc. 1 (1) (1995).

[DG2] T. Drumm, W.M. Goldman, The geometry of crooked planes, Topology 38(1999), p. 323–352.

[F] C. Frances, The conformal boundary of Margulis space-times, C. R. Math.Acad. Sci. Paris 336 (2003), p. 751–756.

[FG] D. Fried, W. M. Goldman, Three-dimensional affine crystallographic groups,Adv. Math. 47 (1983), p. 1–49.

[G] W. M. Goldman, Crooked surfaces and anti-de Sitter geometry, preprint,arXiv:1302.4911.

[GLM1] W. M. Goldman, F. Labourie, G. A. Margulis, Proper affine actions andgeodesic flows of hyperbolic surfaces, Ann. of Math. 170 (2009), p. 1051–1083.

[GLM2] W. M. Goldman, F. Labourie, G. A. Margulis, in preparation.[GLMM] W. M. Goldman, F. Labourie, G. A. Margulis, Y. Minsky, Complete flat

Lorentz 3-manifolds and laminations on hyperbolic surfaces, in preparation.[GM] W. M. Goldman, G. A. Margulis, Flat Lorentz 3-manifolds and cocompact

Fuchsian groups, in Crystallographic groups and their generalizations (Kortrijk,1999), p. 135–145, Contemp. Math. 262, American Mathematical Society, Prov-idence, RI, 2000.

[GK] F. Gueritaud, F. Kassel, Maximally stretched laminations on geometricallyfinite hyperbolic manifolds, preprint, arXiv:1307.0256.

[GKW] F. Gueritaud, F. Kassel, M. Wolff, Compact anti-de Sitter 3-manifolds andfolded hyperbolic structures on surfaces, preprint, arXiv:1311.2804.

[Ka] F. Kassel, Quotients compacts d’espaces homogenes reels ou p-adiques, PhDthesis, Universite Paris-Sud 11, November 2009, available at http://math.

univ-lille1.fr/∼kassel/.[Ke] S. P. Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983),

p. 235–265.[KR] R. S. Kulkarni, F. Raymond, 3-dimensional Lorentz space-forms and Seifert

fiber spaces, J. Differential Geom. 21 (1985), p. 231–268.[Ma1] G. A. Margulis, Free completely discontinuous groups of affine transformations

(in Russian), Dokl. Akad. Nauk SSSR 272 (1983), p. 785–788.[Ma2] G. A. Margulis, Complete affine locally flat manifolds with a free fundamental

group, J. Soviet Math. 1934 (1987), p. 129–139, translated from Zap. Naucha.Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 134 (1984), p. 190–205.

[MM] H. Masur, Y. Minsky, Geometry of the complex of curves. I. Hyperbolicity,Invent. Math. 138 (1999), p. 103–149.

[Me] G. Mess, Lorentz spacetimes of constant curvature (1990), Geom. Dedicata 126(2007), p. 3–45.

[Mi] J. Milnor, On fundamental groups of complete affinely flat manifolds, Adv.Math. 25 (1977), p. 178–187.

[P] R. C. Penner, The decorated Teichmuller space of punctured surfaces, Comm.Math. Phys. 113 (1987), p. 299–339.


[PT] A. Papadopoulos, G. Theret, Shortening all the simple closed geodesics onsurfaces with boundary, Proc. Amer. Math. Soc. 138 (2010), p. 1775–1784.

[T] W. P. Thurston, Minimal stretch maps between hyperbolic surfaces, preprint(1986), arXiv:9801039.

Department of Mathematics, The University of Texas at Austin, 1 Univer-sity Station C1200, Austin, TX 78712, USA

E-mail address: [email protected]

CNRS and Universite Lille 1, Laboratoire Paul Painleve, 59655 Villeneuved’Ascq Cedex, France


CNRS and Universite Lille 1, Laboratoire Paul Painleve, 59655 Villeneuved’Ascq Cedex, France


Documents

MARGULIS SPACETIMES VIA THE ARC COMPLEX - … · margulis spacetimes via the arc complex preliminary version (january 2014) | check back soon for an updated draft. jeffrey danciger,