Quasiconvexity and relatively hyperbolic groups that split

TL;DR

This paper investigates the conditions under which a group splitting as a graph of relatively hyperbolic groups retains relative hyperbolicity and quasiconvexity properties, providing new proofs and criteria.

Contribution

It introduces a new criterion for relative quasiconvexity of subgroups in split groups and offers simplified proofs of relative hyperbolicity using the fine graph approach.

Findings

Short proof of relative hyperbolicity under certain conditions

Criterion for relative quasiconvexity based on intersections with vertex groups

Application to local relative quasiconvexity

Abstract

We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain conditions. We then provide a criterion for the relative quasiconvexity of a subgroup H depending on the relative quasiconvexity of the intersection of H with the vertex groups of G. We give an application towards local relative quasiconvexity.