Topological full groups and t.d.l.c. completions of Thompson's $V$

TL;DR

This paper reinterprets topological full groups from shifts of finite type as groups of colour-preserving tree automorphisms and demonstrates their t.d.l.c. completions with arbitrary finite local prime content, including Thompson's V.

Contribution

It provides a new perspective on topological full groups as tree automorphism groups and constructs their t.d.l.c. completions with specified prime content.

Findings

Topological full groups can be viewed as groups of colour-preserving tree automorphisms.

These groups admit t.d.l.c. completions with arbitrary finite local prime content.

The results apply specifically to Thompson's V group.

Abstract

We show how all topological full groups coming from a one-sided irreducible shift of finite type, as studied by Matui, can be re-interpreted as groups of colour-preserving tree almost automorphisms. As an application, we show that they admit t.d.l.c. completions of arbitrary finite local prime content. This applies in particular to Thompson's $V$.