Quasiconvexity in relatively hyperbolic groups

Abstract

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.