Quasiconvexity in the 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.