The space of directions for hyperbolic totally disconnected locally compact groups

TL;DR

This paper explicitly computes the boundary-like space of directions for certain hyperbolic totally disconnected locally compact groups acting on graphs, confirming a conjecture and extending geometric group theory techniques.

Contribution

It provides the first explicit calculation of the space of directions for these groups, demonstrating its discreteness and generalizing geometric group theory methods to non-discrete cases.

Findings

The space of directions is a discrete metric space.

The results confirm a conjecture by Baumgartner, Möller, and Willis.

Techniques from geometric group theory are extended to non-discrete groups.

Abstract

The space of directions is a notion of boundary associated to an arbitrary totally disconnected locally compact group. We explicitly calculate the space of directions of a group acting vertex transitively with compact open vertex stabilisers on a locally finite connected hyperbolic graph. These are examples of groups where techniques from geometric group theory can be generalised from the discrete to the non-discrete case. We show the space of directions for these groups is a discrete metric space. Our results resolve a conjecture of Baumgartner, M\"oller and Willis in the affirmative.