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2010 Relative hyperbolicity and relative quasiconvexity for countable groups
G Christopher Hruska
Algebr. Geom. Topol. 10(3): 1807-1856 (2010). DOI: 10.2140/agt.2010.10.1807


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.


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G Christopher Hruska. "Relative hyperbolicity and relative quasiconvexity for countable groups." Algebr. Geom. Topol. 10 (3) 1807 - 1856, 2010.


Received: 16 April 2009; Revised: 24 April 2010; Accepted: 10 May 2010; Published: 2010
First available in Project Euclid: 19 December 2017

zbMATH: 1202.20046
MathSciNet: MR2684983
Digital Object Identifier: 10.2140/agt.2010.10.1807

Primary: 20F65, 20F67

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.10 • No. 3 • 2010
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