On combination theorems and Bowditch boundaries of relatively hyperbolic TDLC groups

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.