Bounded subgroups of relatively finitely presented groups

Abstract

Given a finitely generated group G that is relatively finitely presented with respect to a collection of peripheral subgroups, we prove that every infinite subgroup H of G that is bounded in the relative Cayley graph of G is conjugate into a peripheral subgroup. As an application, we obtain a trichotomy for subgroups of relatively hyperbolic groups. Moreover we prove the existence of the relative exponential growth rate for all subgroups of limit groups.