Limit groups for relatively hyperbolic groups, I: The basic tools. Preprint (2008).

Abstract We begin the investigation of G-limit groups, where G is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of Drutu and Sapir, we adapt the results from two previous papers of the author to this context. Specifically, given a finitely generated group H, and a sequence of pairwise non-conjugate homomorphisms { h_n : H --> G}, we extract an R-tree with a nontrivial isometric H-action. We then provide an analogue of Sela's shortening argument. NOTE: This paper was updated in March 2008 to include the results of the previous version of this paper, and also the paper entitled `Limits of (certain) CAT(0) groups II: The shortening argument and the Hopf property'. The previous version of this paper is on the arxiv (in the history of this paper).