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We begin the investigation of Gamma-limit groups, where Gamma is atorsion-free group which is hyperbolic relative to a collection of free abeliansubgroups. Using the results of Drutu and Sapir, we adapt the results frommath.GR/0404440 to this context. Specifically, given a finitely generated groupG, 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.Comment: 41 pages. The new version of this paper has been substantially  rewritten. It now includes all of the results of the previous version, and  also of math.GR/0408080. The exception to this is the proof of the Hopf  property, which follows imediately from Theorem 5.2 of math.GR/0503045 (and  does not use anything omitted from this version

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