Topological full groups of one-sided shifts of finite type

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-infinity, 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.