Pattern rigidity in hyperbolic spaces:  duality and PD subgroups

Biswas, Kingshook; Mj, Mahan

doi:10.4171/ggd/152

Cited by 11 publications

(18 citation statements)

References 35 publications

(74 reference statements)

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“…These papers used an action on a topological space and discreteness of the commensurator stemmed from the fact that the commensurator preserved a 'discrete geometric pattern' (in the sense of Schwartz, cf. [2,14,18,19]). In this paper we use an algebraic action from Chevalley-Weil theory and homological algebra as a replacement in order to conclude discreteness of the commensurator.…”

Section: The Main Resultsmentioning

confidence: 99%

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Commutators, commensurators, and PSL2(Z)

Koberda

2021

Journal of Topology

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Let H < PSL2(Z) be a finite index normal subgroup which is contained in a principal congruence subgroup, and let Φ(H) = H denote a term of the lower central series or the derived series of H. In this paper, we prove that the commensurator of Φ(H) in PSL2(R) is discrete. We thus obtain a natural family of thin subgroups of PSL2(R) whose commensurators are discrete, establishing some cases of a conjecture of Shalom.

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Section: The Main Resultsmentioning

confidence: 99%

“…If Σ has finite genus and only finitely many punctures and no boundary components, then Σ has finite volume and so Γ is not thin. If Σ has a boundary component (that is, a flaring end), then Σ admits a proper geodesically convex core, in which case the limit set Λ Γ will be properly contained in ∂H 2 .…”

Section: Introductionmentioning

confidence: 99%

Commutators, commensurators, and PSL2(Z)

Koberda

2021

Journal of Topology

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“…, H n are cyclic. Biswas-Mj [BM08] generalize this to the case of subgroups which are either codimension one duality groups or odd-dimensional Poincare Duality groups or subgroups containing such subgroups as free factors.…”

Section: Givesmentioning

confidence: 99%

Flows, fixed points and rigidity for Kleinian groups

Biswas

2012

Geom. Funct. Anal.

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Abstract. We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group G 1 and a quasiconformal conjugate h −1 G 2 h of a cocompact group G 2 . We show that if the conjugacy h is not conformal then this group contains a non-trivial one parameter subgroup. This leads to rigidity results; for example, Mostow rigidity is an immediate consequence. We are also able to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group G, by a G-invariant pattern we mean a G-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of G. We prove that (in dimension at least three) for G 1 , G 2 cocompact Kleinian groups, any quasiconformal map pairing a G 1 -invariant pattern to a G 2 -invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas-Mj [BM08] who proved rigidity for Poincare Duality subgroups.

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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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Pattern rigidity in hyperbolic spaces: duality and PD subgroups

Cited by 11 publications

References 35 publications

Commutators, commensurators, and PSL2(Z)

Commutators, commensurators, and PSL2(Z)

Flows, fixed points and rigidity for Kleinian groups

Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces

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