  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.

Group Theory