September 28, 2025Open Access

On combination theorems and Bowditch boundaries of relatively hyperbolic TDLC groups

Puntos clave

Relatively hyperbolic TDLC groups exhibit unique Bowditch boundaries based on compactly generated structures, indicating significant topological features.
When $G$ has infinite rough ends, the Bowditch boundary’s topology is entirely determined by that of its components $A$ and $B$ in the free product decomposition.
Analyzing compactly generated TDLC groups shows their equivalence to previously established concepts of relative hyperbolicity and boundary construction methods.
If a TDLC group has just one rough end, its Bowditch boundary maintains a connected topology, providing insights into group structure.

Resumen

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=AC B or G=AC 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.

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