Relatively quasiconvex subgroups of a countable relatively hyperbolic group
by
Chris Hruska
University of Wisconsin-Milwaukee

Quasiconvex subgroups play a central role in the theory of word hyperbolic groups. The two most fundamental properties of quasiconvex subgroups are the following:

1. A quasiconvex subgroup is itself word hyperbolic.

2. The intersection of two quasiconvex subgroups is quasiconvex.

In this talk, I will introduce the analogous notion of a relatively quasiconvex subgroup H of a countable relatively hyperbolic group G, which satisfies analogues of the two fundamental properties above. This notion can be formulated in several equivalent ways corresponding to the "coned-off" gemetry, the "cusped-off" geometry, and (in the finitely generated case) the word metric. Two of these formulations were introduced by Dahmani and Osin in special cases, but their definitions were not known to be equivalent, and neither was known to satisfy both of the fundamental properties.

In order to obtain both fundamental properties above, we are forced to consider countable relatively hyperbolic groups (not necessarily finitely generated). In fact the theory of relative hyperbolicity is quite natural in this setting, and it incorporates definitions proposed by Gromov, Bowditch, and Osin (which were not previously known to be equivalent).

Paper reference: arXiv:0801.4596

Date received: February 29, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cawj-18.