Flows, fixed points and rigidity for Kleinian groups

Biswas, Kingshook

doi:10.1007/s00039-012-0165-8

Cited by 5 publications

(6 citation statements)

References 9 publications

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“…In the context of hyperbolic geometry, the following notion goes back to Schwartz [53,54] (see also [5,6,42,49] for the connection to rigidity questions). Let 𝐝𝕊 denote ((𝕊 1 × 𝕊 1 ⧵ Δ)∕ ∼), where Δ denotes the diagonal and ∼ denotes the flip equivalence relation.…”

Section: Two-point Characterizations and Modulimentioning

confidence: 99%

Combining rational maps and Kleinian groups via orbit equivalence

Mukherjee

2023

Proceedings of London Math Soc

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We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion‐free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are orbit equivalent to Fuchsian punctured sphere groups. We call these higher Bowen–Series maps. The existence of this class ensures that the Teichmüller space of matings has one component corresponding to Bowen–Series maps and one corresponding to higher Bowen–Series maps. We also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers boundary groups that are mateable in our framework.

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Section: Two-point Characterizations and Modulimentioning

confidence: 99%

Combining rational maps and Kleinian groups via orbit equivalence

Mukherjee

2023

Proceedings of London Math Soc

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“…It is a classical fact that for real hyperbolic space H n , any Moebius map f : ∂H n → ∂H n extends to an isometry F : H n → H n . This fact is a crucial part of many rigidity theorems for hyperbolic spaces, for example the Mostow Rigidity theorem ( [Mos68]) and various "pattern rigidity" theorems ( [Sch97], [BM08], [Bis09]). More generally, Bourdon [Bou96] showed that if X is a rank one symmetric space of noncompact type (with the metric normalized so that the maximum of the sectional curvatures equals −1), and Y is any CAT(-1) space, then any Moebius embedding f : ∂X → ∂Y extends to an isometric embedding F : X → Y .…”

Section: Introductionmentioning

confidence: 99%

Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces

Biswas¹

2021

Preprint

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A quasi-metric antipodal space (Z, ρ 0 ) is a compact space Z with a continuous quasi-metric ρ 0 which is of diameter one, and which is antipodal, i.e. for any ξ ∈ Z there exists η ∈ Z such that ρ 0 (ξ, η) = 1. The quasi-metric ρ 0 defines a positive cross-ratio function on the space of quadruples of distinct points in Z, and a homeomorphism between quasi-metric antipodal spaces is said to be Moebius if it preserves cross-ratios.A proper, geodesically complete Gromov hyperbolic space X is said to be boundary continuous if the Gromov inner product extends continuously to the boundary. Then the boundary ∂X equipped with a visual quasi-metric is a quasi-metric antipodal space. The space X is said to be maximal if for any proper, geodesically complete, boundary continuous Gromov hyperbolic space Y , if there is a Moebius homeomorphism f : ∂Y → ∂X, then f extends to an isometric embedding F : Y → X. We call such spaces maximal Gromov hyperbolic spaces.We show that any proper, geodesically complete, boundary continuous Gromov hyperbolic space embeds isometrically into a maximal Gromov hyperbolic space which is unique up to isometry, and the image of the embedding is cobounded. We prove an equivalence of categories between quasimetric antipodal spaces and maximal Gromov hyperbolic spaces, namely: any quasi-metric antipodal space is Moebius homeomorphic to the boundary of a maximal Gromov hyperbolic space, and any Moebius homeomorphism f : ∂X → ∂Y between boundaries of maximal Gromov hyperbolic spaces X, Y extends to a surjective isometry F : X → Y . This is part of a more general equivalence between certain compact spaces called antipodal spaces and their associated metric spaces called Moebius spaces.

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“…In [BM12], Biswas and Mj generalized Schwartz' result to certain Duality and PD subgroups of rank one symmetric spaces. In [Bis12], Biswas completely solved the pattern rigidity problem for G a uniform lattice in real hyperbolic space and H any infinite quasiconvex subgroup of infinite index in G. However, all these papers used, in an essential way, the linear structure of the groups involved, and the techniques fail for G the fundamental group of a general closed negatively curved manifold. (This point is specifically mentioned by Schwartz in [Sch97]).…”

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confidence: 99%

“…(This point is specifically mentioned by Schwartz in [Sch97]). Further, the study in [Sch95], [Sch97], [BM12], [Bis12] boils down to the study of a single pattern-preserving quasi-isometry between pairs (G 1 , H 1 ) and (G 2 , H 2 ). We propose a different perspective in this paper by studying the full group P P QI(G, H) of pattern-preserving (self) quasi-isometries of a pair (G, H) for G a hyperbolic group and H any infinite quasiconvex subgroup of infinite index.…”

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confidence: 99%

Pattern rigidity and the Hilbert–Smith conjecture

2012

Geom. Topol.

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We initiate a study of the topological group PPQI.G; H / of pattern-preserving quasiisometries for G a hyperbolic Poincaré duality group and H an infinite quasiconvex subgroup of infinite index in G . Suppose @G admits a visual metric d with dim haus < dim t C 2, where dim haus is the Hausdorff dimension and dim t is the topological dimension of .@G; d /. Equivalently suppose that ACD.@G/ < dim t C 2, where ACD.@G/ denotes the Ahlfors regular conformal dimension of @G .(a) If Q u is a group of pattern-preserving uniform quasi-isometries (or more generally any locally compact group of pattern-preserving quasi-isometries) containing G , then G is of finite index in Q u . (b) If instead, H is a codimension one filling subgroup, and Q is any group of pattern-preserving quasi-isometries containing G , then G is of finite index in Q. Moreover, if L is the limit set of H , L is the collection of translates of L under G , and Q is any pattern-preserving group of homeomorphisms of @G preserving L and containing G , then the index of G in Q is finite (Topological Pattern Rigidity).We find analogous results in the realm of relative hyperbolicity, regarding an equivariant collection of horoballs as a symmetric pattern in the universal cover of a complete finite volume noncompact manifold of pinched negative curvature. Our main result combined with a theorem of Mosher, Sageev and Whyte gives QI rigidity results.An important ingredient of the proof is a version of the Hilbert-Smith conjecture for certain metric measure spaces, which uses the full strength of Yang's theorem on actions of the p-adic integers on homology manifolds. This might be of independent interest.

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Flows, fixed points and rigidity for Kleinian groups

Cited by 5 publications

References 9 publications

Combining rational maps and Kleinian groups via orbit equivalence

Combining rational maps and Kleinian groups via orbit equivalence

Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces

Pattern rigidity and the Hilbert–Smith conjecture

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