Limit groups for relatively hyperbolic groups. I. the basic tools

Abstract

We begin the investigation of Γ-limit groups, where Γ is a torsion free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of Druţu and Sapir [Topology 44 (2005) 959 1058], we adapt the results from the author 's paper [Algebr. Geom. Topol. 5 (2005) 1325 1364]. Specifically, given a finitely generated group G and a sequence of pairwise nonconjugate homomorphisms {hn: G Γ}, we extract an R-tree with a nontrivial isometric G-action. We then provide an analogue of Sela 's shortening argument. © 2009 Mathematical Sciences Publishers.