Cayley--Abels graphs and invariants of totally disconnected, locally compact groups

TL;DR

This paper explores Cayley--Abels graphs for totally disconnected, locally compact groups, relating their minimal degree to group invariants and demonstrating that for automorphism groups of regular trees, the minimal degree equals the degree of the tree.

Contribution

It introduces the concept of minimal degree for such groups and connects it to key invariants like the modular and scale functions, providing new insights into their structure.

Findings

Minimal degree of Aut(T_d) equals d for all d ≥ 2.

Established relationships between minimal degree and group invariants.

Demonstrated the minimal degree's significance in understanding group actions.

Abstract

A connected, locally finite graph $\Gamma$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $\Gamma$. Define the minimal degree of $G$ as the minimal degree of a Cayley--Abels graph of $G$. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_d$ denotes the $d$-regular tree, then the minimal degree of ${\rm Aut}(T_d)$ is $d$ for all $d\geq 2$.