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

Groves, Daniel

doi:10.2140/agt.2009.9.1423

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

Daniel Groves

Algebraic & Geometric Topology 9 (2009) 1423–1466

DOI: 10.2140/agt.2009.9.1423

Abstract

We begin the investigation of $Γ$ –limit groups, where $Γ$ is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of Druţu and Sapir [Topology 44 (2005) 959-1058], we adapt the results from the author’s paper [Algebr. Geom. Topol. 5 (2005) 1325-1364]. Specifically, given a finitely generated group $G$ and a sequence of pairwise nonconjugate homomorphisms ${h_{n} : G \to Γ}$ , we extract an $ℝ$ –tree with a nontrivial isometric $G$ –action.

We then provide an analogue of Sela’s shortening argument.

Keywords

relatively hyperbolic group, limit group

Mathematical Subject Classification 2000

Primary: 20F65

Secondary: 20F67, 20E08, 57M07

References

Publication

Received: 20 March 2008

Revised: 17 December 2008

Accepted: 14 May 2009

Published: 26 July 2009

Authors

Daniel Groves
Department of Mathematics University of Illinois at Chicago 851 S Morgan St Chicago, IL 60607-7045 USA
http://www.math.uic.edu/~groves

Volume 9, issue 3 (2009)