Quasiconvexity in the Relatively Hyperbolic Groups
Group Theory
2011-10-12 v4 Geometric Topology
Abstract
We study different notions of quasiconvexity for a subgroup of a relatively hyperbolic group The first result establishes equivalent conditions for 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 \cite{Os06}. In the second part of the paper we prove that a subgroup of a finitely generated relatively hyperbolic group acts cocompactly outside its limit set if and only if it is (absolutely) quasiconvex and every its infinite intersection with a parabolic subgroup of has finite index in the parabolic subgroup. Consequently we obtain a list of different subgroup properties and establish relations between them.
Cite
@article{arxiv.1103.1211,
title = {Quasiconvexity in the Relatively Hyperbolic Groups},
author = {Victor Gerasimov and Leonid Potyagailo},
journal= {arXiv preprint arXiv:1103.1211},
year = {2011}
}