Limit groups for relatively hyperbolic groups, I: The basic tools

Abstract

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.

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)

References in corpus (7)

• Combination of convergence groups
• Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams
• Hadamard spaces with isolated flats
• The Isomorphism Problem for Toral Relatively Hyperbolic Groups
• Accidental parabolics and relatively hyperbolic groups
• Makanin-Razborov diagrams for limit groups
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• A Cartan-Hadamard type result for relatively hyperbolic groups
• On R-trees, homotopies, and covering maps