#### Volume 9, issue 3 (2009)

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

### Daniel Groves

Algebraic & Geometric Topology 9 (2009) 1423–1466
##### 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.

##### Keywords
relatively hyperbolic group, limit group
##### Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20F67, 20E08, 57M07