Topological full groups of one-sided shifts of finite type

TL;DR

This paper studies the topological full groups of one-sided shifts of finite type, showing they are finitely presented, simple in certain cases, and serve as invariants for the underlying groupoids, thus generalizing Higman-Thompson groups.

Contribution

It introduces a new class of finitely presented infinite simple groups derived from one-sided shifts of finite type and establishes their properties and invariants.

Findings

[[G]] is of type F_infinity, hence finitely presented.

The commutator subgroup D([[G]]) is simple for certain groupoids.

[[G]] generalizes Higman-Thompson groups and classifies groupoid isomorphisms.

Abstract

We explore the topological full group [[G]] of an essentially principal etale groupoid G on a Cantor set. When G is minimal, we show that [[G]] (and its certain normal subgroup) is a complete invariant for the isomorphism class of the etale groupoid G. Furthermore, when G is either almost finite or purely infinite, the commutator subgroup D([[G]]) is shown to be simple. The etale groupoid G arising from a one-sided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid. For such G, [[G]] is thought of as a generalization of the Higman-Thompson group. We prove that [[G]] is of type F_\infty, and so in particular it is finitely presented. This gives us a new infinite family of finitely presented infinite simple groups. Also, the abelianization of [[G]] is calculated and described in terms of the homology groups of G.