Conical limit points and uniform convergence groups

Tukia, Pekka

doi:10.1515/crll.1998.081

Cited by 67 publications

(97 citation statements)

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“…Therefore it is not a parabolic point (see the result of Tukia, described in [6] Prop.3.2, see also [31]), and each parabolic point for H in ΛH is a parabolic point for Γ in M , and its maximal parabolic subgroup in H is exactly the intersection of its maximal parabolic subgroup in Γ with H .…”

Section: Remark (I)mentioning

confidence: 99%

Combination of convergence groups

2003

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We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively hyperbolic groups over a fully quasi-convex subgroup. We apply our result to Sela's theory on limit groups and prove their relative hyperbolicity. We also get a proof of the Howson property for limit groups.

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Section: Remark (I)mentioning

confidence: 99%

Combination of convergence groups

2003

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“…Indeed in this case by [Ge1,Main Theorem.c] the space T is metrisable. So by [Tu3] the action G T is 2-cocompact and the above Corollary holds.…”

Section: ) Otherwise B D and We Havementioning

confidence: 81%

“…Indeed the sufficiency follows from P. Tukia's result [Tu3,Theorem 1.C]. The converse statement is a partial case of [Ge1,Main Theorem,b].…”

Section: )mentioning

confidence: 93%

“…every point of T is either conical or bounded parabolic or isolated [Ge1]. Conversely if a group G admits a minimal geometrically finite action on a metrisable space T without isolated points then the action is 2-cocompact [Tu3]. If G is finitely generated then the existence of a geometrically finite action of G is equivalent (see [Bo1] and [Ya]) to the "classical" relative hyperbolicity in the sense of Farb [Fa] and Gromov [Gr,8.6].…”

Section: Introductionmentioning

confidence: 99%

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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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“…It follows from [Ge09], [Tu98] that an action of a finitely generated group G on a compactum X is geometrically finite if and only if it admits a 3-discontinuous and 2-cocompact action on X. So we will further regard the existence of a 3-discontinuous and 2-cocompact action as the definition of RHG.…”

mentioning

confidence: 99%

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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Conical limit points and uniform convergence groups

Cited by 67 publications

References 0 publications

Combination of convergence groups

Combination of convergence groups

Non-finitely generated relatively hyperbolic groups and Floyd quasiconvexity

Quasiconvexity in relatively hyperbolic groups

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