Quasiconvexity in relatively hyperbolic groups
-
Victor Gerasimov
and
Leonid Potyagailo
Abstract
We study different notions of quasiconvexity for a subgroup H of a relatively hyperbolic group G. Our first result implies that relative geometric quasiconvexity is equivalent to dynamical quasiconvexity as it was conjectured by D. Osin. 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 infinite intersection of H with a parabolic subgroup of G has finite index in the parabolic subgroup. We also discuss relations between other properties of subgroups close to quasiconvexity.
Funding source: ANR
Award Identifier / Grant number: BLAN 2011 BS0101304
The authors wish to thank Misha Kapovich and Wenyuan Yang for useful discussions and suggestions. They also gratefully acknowledge the many useful corrections and remarks of the anonymous referee. The second author is thankful to the Mittag-Leffler Institute for the hospitality during a semester “Geometric and Analytic Aspects of Group Theory”, and to the Max-Planck Institute of Mathematics in Bonn where an essential part of this work has been done.
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- The Minkowski problem, new constant curvature surfaces in ℝ3, and some applications
- The asymptotic expansion of a hypergeometric series coming from mirror symmetry
- Convergence of metrics under self-dual Weyl tensor and scalar curvature bounds
- Bounded sets of sheaves on Kähler manifolds
- Quasiconvexity in relatively hyperbolic groups
- Twisted Hilbert transforms vs Kakeya sets of directions
- Approximation forte en famille
- Arbitrarily large residual finiteness growth
- Big denominators and analytic normal forms
Articles in the same Issue
- Frontmatter
- The Minkowski problem, new constant curvature surfaces in ℝ3, and some applications
- The asymptotic expansion of a hypergeometric series coming from mirror symmetry
- Convergence of metrics under self-dual Weyl tensor and scalar curvature bounds
- Bounded sets of sheaves on Kähler manifolds
- Quasiconvexity in relatively hyperbolic groups
- Twisted Hilbert transforms vs Kakeya sets of directions
- Approximation forte en famille
- Arbitrarily large residual finiteness growth
- Big denominators and analytic normal forms
