Quasiconvexity in the Relatively Hyperbolic Groups

TL;DR

This paper investigates various notions of quasiconvexity in relatively hyperbolic groups, establishing equivalences and characterizations that connect geometric, dynamical, and algebraic properties of subgroups.

Contribution

It provides new equivalences for relative quasiconvexity and characterizes subgroups acting cocompactly outside their limit sets, answering a question by D. Osin.

Findings

Relative quasiconvexity is equivalent to dynamical quasiconvexity.

Subgroups acting cocompactly outside their limit set are exactly those that are quasiconvex with finite index intersections with parabolic subgroups.

The paper establishes relations between different subgroup properties in relatively hyperbolic groups.

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 \cite{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. Consequently we obtain a list of different subgroup properties and establish relations between them.