On Quasiconvexity and Relative Hyperbolic Structures

TL;DR

This paper explores the relationships between different relative hyperbolic structures of a group, specifically how quasiconvexity properties transfer between collections of subgroups, with theoretical characterizations and applications.

Contribution

It provides a characterization of when relative quasiconvexity with respect to one collection implies quasiconvexity with respect to a larger collection, and vice versa.

Findings

Quasiconvexity relative to A implies quasiconvexity relative to B.

Quasiconvexity relative to B implies quasiconvexity relative to A.

Theoretical applications in group hyperbolic structures.

Abstract

Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when quasicovexity relative to A implies quasiconvexity relative to B. We also show that quasiconvexity relative to B implies quasiconvexity relative to A. Some applications are discussed.