Topological full groups of one-sided shifts of finite type

Abstract We explore the topological full group 〚 G 〛 G of an essentially principal étale groupoid G on a Cantor set. When G is minimal, we show that 〚 G 〛 G (and its certain normal subgroup) is a complete invariant for the isomorphism class of the étale groupoid G. Furthermore, when G is either almost finite or purely infinite, the commutator subgroup D (〚 G 〛) D (G) is shown to be simple. The étale 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 〛 G is thought of as a generalization of the Higman–Thompson group. We prove that 〚 G 〛 G is of type F ∞, 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 〛 G is calculated and described in terms of the homology groups of G.