Limit groups for relatively hyperbolic groups, I: The basic tools. Preprint (2004).

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. This, along with the analogue of Sela's shortening argument allows us to prove the main result of this paper, that G is Hopfian.