Simple locally compact groups acting on trees and their germs of automorphisms

TL;DR

This paper investigates the structure of automorphism groups of locally finite trees, focusing on the relationship between local and global properties, and introduces new examples of simple, compactly generated groups related to Neretin's spheromorphisms.

Contribution

It characterizes when such groups have few open subgroups, explores automorphism germs, and computes the full group of germs, extending understanding of self-replicating profinite branch groups.

Findings

Groups with local primitivity have few open subgroups

Existence of many automorphism germs not extending to global automorphisms

Construction of new simple, compactly generated groups generalizing Neretin's group

Abstract

Automorphism groups of locally finite trees provide a large class of examples of simple totally disconnected locally compact groups. It is desirable to understand the connections between the global and local structure of such a group. Topologically, the local structure is given by the commensurability class of a vertex stabiliser; on the other hand, the action on the tree suggests that the local structure should correspond to the local action of a stabiliser of a vertex on its neighbours. We study the interplay between these different aspects for the special class of groups satisfying Tits' independence property. We show that such a group has few open subgroups if and only if it acts locally primitively. Moreover, we show that it always admits many germs of automorphisms which do not extend to automorphisms, from which we deduce a negative answer to a question by George Willis. Finally, under suitable assumptions, we compute the full group of germs of automorphisms; in some specific cases, these turn out to be simple and compactly generated, thereby providing a new infinite family of examples which generalise Neretin's group of spheromorphisms. Our methods describe more generally the abstract commensurator group for a large family of self-replicating profinite branch groups.