Relative Quasiconvexity using Fine Hyperbolic Graphs

TL;DR

This paper introduces a new approach to relative quasiconvexity in relatively hyperbolic groups using fine hyperbolic graphs, generalizing previous definitions and proving that quasiconvex subgroups are relatively hyperbolic.

Contribution

It presents an elegant, generalized framework for relative quasiconvexity applicable to uncountable groups and offers a self-contained proof of the hyperbolicity of quasiconvex subgroups.

Findings

Generalizes existing definitions to uncountable groups

Provides an elementary proof that quasiconvex subgroups are relatively hyperbolic

Uses Bowditch's approach with cocompact actions on fine hyperbolic graphs

Abstract

We provide a new and elegant approach to relative quasiconvexity for relatively hyperbolic groups in the context of Bowditch's approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach to quasiconvexity generalizes the other definitions in the literature that apply only for countable relatively hyperbolic groups. We also provide an elementary and self-contained proof that relatively quasiconvex subgroups are relatively hyperbolic.