Topological full groups of one-sided shifts of finite type

We explore the topological full group of an essentially principal étale groupoid on a Cantor set. When is minimal, we show that (and its certain normal subgroup) is a complete invariant for the isomorphism class of the étale groupoid . Furthermore, when is either almost finite or purely infinite, the commutator subgroup is shown to be simple. The étale groupoid arising from a one-sided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid. For such , is thought of as a generalization of the Higman–Thompson group. We prove that is of type , 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 is calculated and described in terms of the homology groups of .