A combination theorem for relatively hyperbolic groups and finite relative height of splitting

TL;DR

This paper establishes a combination theorem for relatively hyperbolic groups in graphs of groups, linking relative quasi-convexity of edge groups to their finite relative height, and extends boundary construction techniques.

Contribution

It introduces a new combination theorem for relatively hyperbolic groups with acylindrical graphs, extending previous boundary construction methods and characterizing edge groups' properties.

Findings

Edge groups have finite relative height iff they are relatively quasi-convex.

Theorem extends boundary construction techniques to new group configurations.

Provides an application demonstrating the theorem's utility.

Abstract

In this paper, we prove a combination theorem for a relatively acylindrical graph of relatively hyperbolic groups (Theorem 1.1). Here, we are extending the technique of [Tom21] and constructing Bowditch boundary of the fundamental group of graph of groups. Suppose G(Y) is a graph of relatively hyperbolic groups such that edge groups are relatively quasi-convex in adjacent vertex groups. Also, assume that the fundamental group of G(Y) is relatively hyperbolic. Then we show that the edge groups of G(Y) have finite relative height (Definition 1.5) if and only if they are relatively quasi-convex (Theorem 1.6). In the last section, we give an application.