Relative hyperbolicity and relative quasiconvexity for countable groups

TL;DR

This paper develops a foundational theory for relatively quasiconvex subgroups within countable relatively hyperbolic groups, establishing equivalences of definitions and properties of these subgroups.

Contribution

It introduces a coherent framework for countable relatively hyperbolic groups and characterizes relatively quasiconvex subgroups, including their stability under intersection and their hyperbolicity.

Findings

Proves equivalence of different definitions of relative hyperbolicity for countable groups.

Shows that relatively quasiconvex subgroups are themselves relatively hyperbolic.

Demonstrates that intersections of relatively quasiconvex subgroups are relatively quasiconvex.

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.