Quasiconvexity in the Relatively Hyperbolic Groups

2011/03/07 by Victor Gerasimov, Gerasimov, Victor, Leonid Potyagailo +1 · 1 citation
Mathematics · #20F65 (Primary) 20F67 #57M07 (Secondary) 30F40 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20F65 #msc:20F67 #msc:30F40 #msc:57M07

paper · pdf · doi:10.48550/arxiv.1103.1211

arxiv created 2011/10/11 · arxiv updated 2011/10/12

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 \citeOs06. 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.

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