On combination theorems and Bowditch boundaries of relatively hyperbolic TDLC groups

TL;DR

This paper extends the theory of relatively hyperbolic groups to totally disconnected locally compact groups, introduces a new approach, and explores the properties of their Bowditch boundaries, including connectedness and boundary topology.

Contribution

It introduces an approach to relative hyperbolicity for TDLC groups, proves equivalence with existing notions, and analyzes boundary properties for group splittings.

Findings

G is relatively hyperbolic if A and B are relatively hyperbolic TDLC groups and C is compact.

The topology of G's Bowditch boundary is determined by A and B's boundaries when G has infinite rough ends.

If G has one rough end, its Bowditch boundary is connected.

Abstract

Based on the work of Farb, Bowditch, and Groves-Manning on discrete relatively hyperbolic groups, we introduce an approach to relative hyperbolicity for totally disconnected locally compact (TDLC) groups. For compactly generated TDLC groups, we prove that this notion is equivalent to the one introduced by Arora-Pedroza. Let $G=A\ast_C B$ or $G=A\ast_C$ where $A$ and $B$ are relatively hyperbolic TDLC groups and $C$ is compact. We prove that $G$ is a relatively hyperbolic TDLC group and give a construction of the Bowditch boundary of $G$. As a consequence, we prove that if the rough ends of $G$ are infinite, then the topology of the Bowditch boundary of $G$ is uniquely determined by the topology of the Bowditch boundary of $A$ and $B$. Further, we show that if a relatively hyperbolic TDLC group has one rough end, then its Bowditch boundary is connected. Finally, we show that if the Gromov boundary of a hyperbolic TDLC group $G$ is totally disconnected, then $G$ splits as a finite graph of compact groups.