Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

Gerasimov, Victor; Potyagailo, Leonid

doi:10.4171/jems/417

Cited by 46 publications

(88 citation statements)

References 26 publications

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“…Notations and definitions. We keep some notations and terminology of [GP09] and [Ge10]. The canonical distance function on the set Γ 0 of vertices of a graph Γ is denoted by d. By Γ 1 we denote the set of pairs of vertices joined by edges.…”

Section: Karlsson Functions For Generalized Quasigeodesicsmentioning

confidence: 99%

“…Such subset exists if and only if G is finitely generated (see e.g. [GP09] or [GP10] where A is the vertex set of the Cayley graph of G or a G-orbit of an entourage of T ).…”

Section: Distorted Pathsmentioning

confidence: 99%

“…The Floyd map induces a set of shortcut metrics δ v on T (v∈A), where every δ v is the maximal among all metrics ρ on T [GP09].…”

Section: Conventionmentioning

confidence: 99%

“…Generalizing the ideas of [GP09] we prove in Section 4 that the (α-)convex hull of a closed set in X is itself closed in X (4.1.3). As a corollary we obtain that the subgroups of relatively hyperbolic groups acting cocompactly on X outside their limit sets are undistorted (Corollary 4.2.1) and α-quasiconvex (Proposition 4.2).…”

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confidence: 99%

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Quasiconvexity in relatively hyperbolic groups

Gerasimov

Potyagailo

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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ABSTRACT. We study different notions of quasiconvexity for a subgroup H of a relatively hyperbolic group G. The first result establishes equivalent conditions for H to be relatively quasiconvex. As a corollary we obtain that the relative quasiconvexity is equivalent to the dynamical quasiconvexity. This answers to a question posed by D. Osin [Os06].In the second part of the paper we prove that a subgroup H of a finitely generated relatively hyperbolic group G acts cocompactly outside its limit set if and only if it is (absolutely) quasiconvex and every its infinite intersection with a parabolic subgroup of G has finite index in the parabolic subgroup.We then obtain a list of different subgroup properties and establish relations between them.

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Section: Karlsson Functions For Generalized Quasigeodesicsmentioning

confidence: 99%

“…Such subset exists if and only if G is finitely generated (see e.g. [GP09] or [GP10] where A is the vertex set of the Cayley graph of G or a G-orbit of an entourage of T ).…”

Section: Distorted Pathsmentioning

confidence: 99%

“…The Floyd map induces a set of shortcut metrics δ v on T (v∈A), where every δ v is the maximal among all metrics ρ on T [GP09].…”

Section: Conventionmentioning

confidence: 99%

mentioning

confidence: 99%

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Quasiconvexity in relatively hyperbolic groups

Gerasimov

Potyagailo

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In [GePo1,Appendix] we gave a short proof of Bowditch's theorem that the existence of a 3-discontinuous and 3-cocompact action of a finitely generated group implies that the group is hyperbolic. The above Corollary omits the assumption of finite generatedness.…”

Section: Introductionmentioning

confidence: 99%

Non-finitely generated relatively hyperbolic groups and Floyd quasiconvexity

Gerasimov¹,

Potyagailo²

2015

Groups Geom. Dyn.

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Abstract. We regard a relatively hyperbolic group as a group acting non-trivially by homeomorphisms on a compactum T discontinuously on the set of distinct triples and cocompactly on the set of distinct pairs of points of T .In the first part of the paper we prove that such a group G admits a graph of groups decomposition given by a star graph whose central vertex group is finitely generated relatively hyperbolic with respect to the edge groups, and the other vertex groups are stabilizers of non-equivalent parabolic points. It follows from this result that every relatively hyperbolic group is relatively finitely generated with respect to the parabolic subgroups. Another corollary is that the definition of the relative hyperbolicity which we are using is equivalent to those of Bowditch and Osin (taken with respect to finitely many peripheral subgroups) and they are all equivalent to the existence of the above star graph of groups decomposition.The second part of the paper uses the method of the first part. Considering the induced action of G on the space of distinct pairs of T we construct a connected graph on which G acts properly and cofinitely on edges. Equipping the graph with Floyd metrics we prove that the quasigeodesics in this metric are close somewhere to the geodesics in the word metric. This allows us to prove that the parabolic subgroups of G are quasiconvex with respect to the Floyd metrics. As a corollary we prove that the preimage of a parabolic point by the Floyd map is the Floyd boundary of its stabilizer.

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Floyd maps for relatively hyperbolic groups

Gerasimov

2012

Geom. Funct. Anal.

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ABSTRACT. Let δ S,λ denote the Floyd metric on a discrete group G generated by a finite set S with respect to the scaling function fn=λ n for a positive λ<1. We prove that if G is relatively hyperbolic with respect to a collection P of subgroups then there exists λ such that the identity map G → G extends to a continuous equivariant map from the completion with respect to δ S,λ to the Bowditch completion of G with respect to P.In order to optimize the proof and the usage of the map theorem we propose two new definitions of relative hyperbolicity equivalent to the other known definitions.In our approach some "visibility" conditions in graphs are essential. We introduce a class of "visibility actions" that contains the class of relatively hyperbolic actions. The convergence property still holds for the visibility actions.Let a locally compact group G act on a compactum Λ with convergence property and on a locally compact Hausdorff space Ω properly and cocomactly. Then the topologies on Λ and Ω extend uniquely to a topology on the direct union T = Λ⊔Ω making T a compact Hausdorff space such that the action G T has convergence property. We call T the attractor sum of Λ and Ω.

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Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups

Cited by 46 publications

References 26 publications

Quasiconvexity in relatively hyperbolic groups

Quasiconvexity in relatively hyperbolic groups

Non-finitely generated relatively hyperbolic groups and Floyd quasiconvexity

Floyd maps for relatively hyperbolic groups

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