CRITICAL EXPONENTS AND RIGIDITY IN
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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.
Voir
- fuchsian representation
- hausdorff dimension
- invariant convex
- dimensional sphere
- negative curvature
- convex cocompact
- representation ?
- mental group
- dimensional topological
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.
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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∗CBof its subgroupsAandBover the amalgamated subgroupCsuch thatCcocompactly preserves a totally geodesic copy of the hyperbolic spaceHn−1inHn. For such a group Γ the quotient manifoldM=Hn/Γ is a compact hyperbolic manifold with a totally geodesic embedded and separating hypersurfaceN=Hn−1/C can consider a Fuchsian. One representationρ0: 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 spaceHninHm. Let Γ be a lattice ofP O(n,1), a fuchsian representationρ0of Γ inP O(m,1) withm > ncan be obtained by this way:ρ0:A∈Γ→A0I0d∈ P O(m,1).
For such a fuchsian representationρ0of Γ =A∗CBinP O(n+ 1,1) the groupρ0(C) preserves a totally geodesic copy of the hyperbolic space Hn−1inHn+1. The groupρ0(C) is then centralized inP O(n+ 1,1) by the subgroup of rotations aroundHn−1inHn+1which is isomorphic to S1. Forrt=eit∈S1, let us defineρt: Γ→P O(n+ 1,1) byρt(a) =afor alla∈Aandρt(b) =rt−1brtfor allb∈B. Asrtcommutes withρ0(C) there is no ambiguity in the definition ofρt(c) forc∈C=A∩B. It can be shown that fort6= 0 small enough, the groupρ0(Γ) does not preserve any totally geodesic copy ofHninHn+1and thus cannot be conjugate to ρ0, 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
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G=:ρ(Γ) for a non fuchsian representationρis stricly larger than the size of the limit set ofG0=:ρ0(Γ) for any Fuchsian representationρ0. 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) = GoX∪∂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 ifρ0: Γ→P O(n+ 1,1) is a Fuchsian representation of a cocompact lattice Γ ofP O(n,1), thenG0=ρ0(Γ) is a convex cocompact group of the hyperbolic spaceHn+1.The limit set Λ(G0) ofGis the boundary∂Hn, the convex hullH(G0) is the totally geodesic copy of HninHn+1preserved byG0and the convex cocompactness ofG0comes from the cocompactness of Γ. IfGt=ρt(Γ) are bendings then theGt’s are convex cocompact fortsmall enough, and the limit set Λ(Gt) of each suchGtis 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,ξ0be two points in∂X and denote byl(ξ, ξ0) the distance betweenoand the geodesic joiningξ
