CRITICAL EXPONENTS AND RIGIDITY IN

Gilles Courtois

Description

CRITICAL EXPONENTS AND RIGIDITY IN NEGATIVE CURVATURE by Gilles COURTOIS Abstract. — The goal of this lecture is to describe a theorem of M.Bonk and B.Kleiner on the rigidity of discrete groups acting on CAT(- 1)-spaces whose limit set's Hausdorff and topological dimensions coin- cide. We will give the proof of M.Bonk and B.Kleiner and also an alter- native proof in a particular case. Resume (Exposants critiques et rigidite en courbure negative) Dans ces notes nous presentons un theoreme de M. Bonk et B. Kleiner concernant la rigidite des groupes discrets d'isometries sur des espaces CAT(-1) dont les dimensions de Hausdorff et topologiques sont egales. Nous decrivons la preuve de M. Bonk et B. Kleiner ainsi qu'une preuve differente dans un cas particulier. Contents 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Alternative proof of the theorem 1.5 in a simpler case 7 3. A theorem of P. Tukia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4. Weak tangent and self similarity of limit sets.

• fuchsian representation


• hausdorff dimension


• invariant convex


• dimensional sphere


• negative curvature


• convex cocompact


• representation ?


• mental group


• dimensional topological

Sujets

• Bourdon
• Courtois
• Hausdorff dimension
• Curvature

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 23
• Langue: English

Extrait

CRITICAL EXPONENTS AND RIGIDITY NEGATIVE CURVATURE

by

Gilles COURTOIS

IN

Abstract. —The goal of this lecture is to describe a theorem of M.Bonk and B.Kleiner on the rigidity of discrete groups acting on CAT(- 1)-spaces whose limit set’s Hausdorff and topological dimensions coin- cide. We will give the proof of M.Bonk and B.Kleiner and also an alter- native proof in a particular case.

Re´sum´ercstnasoeseuqitixp(E´ngeuber)etaviidittrigcour´een Danscesnotesnouspr´esentonsunthe´or`emedeM.BonketB.Kleiner concernantlarigidite´desgroupesdiscretsd’isom´etriessurdesespaces CAT(-1)dontlesdimensionsdeHausdorffettopologiquessonte´gales. Nousde´crivonslapreuvedeM.BonketB.Kleinerainsiqu’unepreuve diffe´rentedansuncasparticulier.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Alternative proof of the theorem 1.5 in a simpler case 7 3. A theorem of P. Tukia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4. Weak tangent and self similarity of limit sets . . . . . . . . . 15 5. Topological dimension and regular maps . . . . . . . . . . . . . 22 6. Topological dimension and Hausdorff dimension . . . . . . 28 7. Proof of the theorem 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2000Mathematics Subject Classification. —53C24, 53C21. Key words and phrases. —Hausdorff dimension, topological dimension, critical exponent, rigidity.

2

GILLES COURTOIS

1. Introduction

A famous theorem of G.D.Mostow states that a compact hyperbolic manifold of dimensionn≥3 is determined up to isometry by its funda- mental group. In other words, if Γ is a cocompact lattice inP O(n,1), withn≥3, there is a unique faithfull and discrete representationρ: Γ→P O(n,1) up to conjugacy.

On the other hand, for some lattices Γ ofP O(n,1) there exist many faithfull discrete nonconjugate representationsρ: Γ→P O(m,1) 2≤ n < mas described in the following example.

Bendings:Let us assume that a lattice Γ inP O(n,1) is a free product A∗ C Bof its subgroupsAandBover the amalgamated subgroupCsuch thatCcocompactly preserves a totally geodesic copy of the hyperbolic spaceH n−1 inH n . For such a group Γ the quotient manifoldM=H n /Γ is a compact hyperbolic manifold with a totally geodesic embedded and separating hypersurfaceN=H n−1 /C can consider a Fuchsian. One representationρ 0 : Γ→P O(n+ 1, representation1). Aρof a lattice Γ ofP O(n,1) inP O(m,1) with 2≤n < mis called fuchsian ifρ(Γ) preserves a totally geodesic copy of the hyperbolic spaceH n inH m . Let Γ be a lattice ofP O(n,1), a fuchsian representationρ 0 of Γ inP O(m,1) withm > ncan be obtained by this way:ρ 0 :A∈Γ→  A 0I 0 d  ∈ P O(m,1).

For such a fuchsian representationρ 0 of Γ =A∗ C BinP O(n+ 1,1) the groupρ 0 (C) preserves a totally geodesic copy of the hyperbolic space H n−1 inH n+1 . The groupρ 0 (C) is then centralized inP O(n+ 1,1) by the subgroup of rotations aroundH n−1 inH n+1 which is isomorphic to S 1 . Forr t =e it ∈S 1 , let us defineρ t : Γ→P O(n+ 1,1) byρ t (a) =afor alla∈Aandρ t (b) =r t −1 br t for allb∈B. Asr t commutes withρ 0 (C) there is no ambiguity in the definition ofρ t (c) forc∈C=A∩B. It can be shown that fort6= 0 small enough, the groupρ 0 (Γ) does not preserve any totally geodesic copy ofH n inH n+1 and thus cannot be conjugate to ρ 0 , cf. [11]. One way of distinguishing between a Fuchsian and a non Fuchsian representationρof a cocompact lattice Γ ofP O(n,1) intoP O(m,1), m > nis to compare their limit set. the size of the limit set of Basically

CRITICAL EXPONENTS AND RIGIDITY

3

G=:ρ(Γ) for a non fuchsian representationρis stricly larger than the size of the limit set ofG 0 =:ρ 0 (Γ) for any Fuchsian representationρ 0 . Before going further, let us turn to a more general setting and introduce some notations.

LetXbe a CAT(-1)-space, cf. [4]. Examples of CAT(-1)-space are Cartan Hadamard manifold of negative curvatureK≤ − simply1, ie. connected manifolds of negative sectional curvatureK≤ −1. For a discrete group of isometryGof a CAT(-1)-spaceX, we define the limit set Λ(G) ofGas the closure of the orbit of some (and hence any) pointo∈Xin the ideal boundary∂XofX, namely Λ(G) = Go X∪∂X ∩∂X convex hull of Λ(. TheG) is the smallestG-invariant convex subset ofX∪∂Xcontaining Λ(G), and we denote it byH(G). A discrete group of isometryGofXis convex cocompact ifH(G)/Gis a compact subset ofX/G. The convex cocompactness is equivalent to the quasi-convex cocompactness that we define now. A subsetY⊂Xis said quasi-convex if there is a constantC >0 such that every geodesic segment with endpoints inYlies in theC- neighborhood ofY.

Definition 1.1. —LetXbe a CAT(-1)-space andGa discrete group of isometries ofX group. TheGis said quasi-convex cocompact if there exist aG-invariant quasi-convex subsetY⊂Xwith compact quotient Y /G.

For example ifρ 0 : Γ→P O(n+ 1,1) is a Fuchsian representation of a cocompact lattice Γ ofP O(n,1), thenG 0 =ρ 0 (Γ) is a convex cocompact group of the hyperbolic spaceH n+1 .The limit set Λ(G 0 ) ofGis the boundary∂H n , the convex hullH(G 0 ) is the totally geodesic copy of H n inH n+1 preserved byG 0 and the convex cocompactness ofG 0 comes from the cocompactness of Γ. IfG t =ρ t (Γ) are bendings then theG t ’s are convex cocompact fortsmall enough, and the limit set Λ(G t ) of each suchG t is then a topologicaln−1-dimensional sphere [17], [8]. For a CAT(-1) spaceX, let us define a distance on the ideal boundary as follows. Letobe a fixed point inX. Letξ,ξ 0 be two points in∂X and denote byl(ξ, ξ 0 ) the distance betweenoand the geodesic joiningξ