Relative hyperbolicity and relative quasiconvexity for countable groups

Hruska, G Christopher

doi:10.2140/agt.2010.10.1807

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Relative hyperbolicity and relative quasiconvexity for countable groups

G Christopher Hruska

Algebraic & Geometric Topology 10 (2010) 1807–1856

DOI: 10.2140/agt.2010.10.1807

Abstract

We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely generated). We prove the equivalence of Gromov, Osin and Bowditch’s definitions of relative hyperbolicity for countable groups.

We then give several equivalent definitions of relatively quasiconvex subgroups in terms of various natural geometries on a relatively hyperbolic group. We show that each relatively quasiconvex subgroup is itself relatively hyperbolic, and that the intersection of two relatively quasiconvex subgroups is again relatively quasiconvex. In the finitely generated case, we prove that every undistorted subgroup is relatively quasiconvex, and we compute the distortion of a finitely generated relatively quasiconvex subgroup.

Keywords

relative hyperbolicity, quasiconvex

Mathematical Subject Classification 2000

Primary: 20F65, 20F67

References

Publication

Received: 16 April 2009

Revised: 24 April 2010

Accepted: 10 May 2010

Published: 3 September 2010

Authors

G Christopher Hruska
Department of Mathematical Sciences University of Wisconsin–Milwaukee PO Box 413 Milwaukee, WI 53201 USA
http://www.uwm.edu/~chruska

Volume 10, issue 3 (2010)