CAYLEY-ABELS GRAPHS AND INVARIANTS OF TOTALLY DISCONNECTED, LOCALLY COMPACT GROUPS

A connected, locally finite graph Gamma is a Cayley-Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively on Gamma with compact, open vertex stabilizers. 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 Aut(T-d) is d for all d >= 2.