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 Drutu and Sapir, we adapt the results from math.GR/0404440 to this context. Specifically, given a finitely generated group G, and a sequence of pairwise non-conjugate homomorphisms { h_n : G -> Gamma}, we extract an R-tree with a nontrivial isometric G-action. We then prove an analogue of Sela's shortening argument.
Categories: math.GR