Boundaries of graphs of relatively hyperbolic groups with cyclic edge groups

TL;DR

This paper investigates the boundaries of graphs of relatively hyperbolic groups, establishing conditions under which their fundamental groups are relatively hyperbolic and analyzing the topological structure of their Bowditch boundaries.

Contribution

It proves that the fundamental group of a graph of convergence groups with parabolic edge groups is a convergence group and extends this to relatively hyperbolic groups with cyclic edge groups, including boundary homeomorphism results.

Findings

Fundamental groups of certain graphs of convergence groups are convergence groups.

Fundamental groups of graphs of relatively hyperbolic groups with cyclic edge groups are relatively hyperbolic.

The Bowditch boundary's homeomorphism type is determined by vertex groups' boundaries under specific conditions.

Abstract

We prove that the fundamental group of a finite graph of convergence groups with parabolic edge groups is a convergence group. Using this result, under some mild assumptions, we prove a combination theorem for a graph of convergence groups with dynamically quasi-convex edge groups (Theorem 1.3). To prove these results, we use a modification of Dahmani's technique [Dah03]. Then we show that the fundamental group of a graph of relatively hyperbolic groups with edge groups either parabolic or infinite cyclic is relatively hyperbolic and construct Bowditch boundary. Finally, we show that the homeomorphism type of Bowditch boundary of the fundamental group of a graph of relatively hyperbolic groups with parabolic edge groups is determined by the homeomorphism type of the Bowditch boundaries of vertex groups (under some additional hypotheses)(Theorem 7.1). In the last section of the paper, we give some applications and examples.