Height in Splitting of Relatively Hyperbolic Groups

TL;DR

This paper establishes a criterion linking the relative quasiconvexity of vertex groups in a graph of relatively hyperbolic groups to their finite relative height in the fundamental group, advancing understanding of group splittings.

Contribution

It provides a new characterization of vertex groups' relative quasiconvexity based on their relative height within the fundamental group of a graph of relatively hyperbolic groups.

Findings

Vertex groups are relatively quasiconvex iff they have finite relative height in the fundamental group.

The paper proves an equivalence relating geometric properties of groups to algebraic height conditions.

It extends the theory of group splittings in the context of relatively hyperbolic groups.

Abstract

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively quasiconvex if and only if all the vertex groups have finite relative height in the fundamental group.