Connections Between Willis' Theory for Totally Disconnected Locally Compact Groups and Graph Automorphisms

Abstract

We investigate the tidy subgroups, scale function and related invariants of totally disconnected locally compact groups. Our focus is on relating these ideas to combinatorial and geometric aspects of the group. After giving necessary background, we study the scale function and tidy subgroups of an endomorphism of a totally disconnected locally compact group. Our results are inspired by a similar investigation for automorphisms by Möller (Can. J. Math., 54(4), 795-827). We characterise when a compact open subgroup is tidy for an endomorphism in terms of a graph constructed from the subgroups and the endomorphism. Using this characterisation, we develop a tidying procedure which produces from a compact open subgroup, a tidy subgroup. We also use our characterisation to prove a tree representation theorem for endomorphisms, inspired by a similar theorem of Baumgartner and Willis (Isr. J. Math., 142(1), 221-248) for automorphisms. We then study restricted Burger-Mozes groups. These are algebraic subgroups of the autmorphism group of a regular tree but are not equipped with the permutation topology. The stabiliser of a vertex in these groups is open but not compact. We calculate invariants for these groups and relate them to similar calculations done for the automorphism group of a regular tree. This gives insight on how results for the automorphism group of a regular tree may generalise to a larger class of totally disconnected locally compact groups. We investigate the space of directions for a totally disconnected locally compact group acting vertex transitively with compact open vertex stabilisers on a hyperbolic graph. Generalising the discrete case, we call such groups hyperbolic. We show that the space of directions for a hyperbolic group is a discrete metric space and that asymptotic classes are determined by fixed points on the boundary of the hyperbolic graph. This verifies a conjecture of Baumgartner, Möller and Willis (Isr. J. Math., 190(1), 365-388).