Limit groups for relatively hyperbolic groups. {I}. The basic tools

Abstract

We begin the investigation of $\Gamma$–limit groups, where $\Gamma$ 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 $\left\{h_n:G\to \Gamma \right\}$, we extract an $ℝ$–tree with a nontrivial isometric $G$–action.

We then provide an analogue of Sela’s shortening argument.