Topological full groups of one-sided shifts of finite type

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.

52 pages; corrected a mistake in Section 6.7

References in corpus (4)

• Full Groups and Orbit Equivalence in Cantor Dynamics
• Some remarks on topological full groups of Cantor minimal systems II
• Cartan subalgebras in C*-algebras
• Full groups of one-sided topological Markov shifts

Cited by in corpus (11)

• The group of reversible Turing machines
• Finitely presented groups associated with expanding maps
• Amenable minimal Cantor systems of free groups arising from diagonal actions
• Construction of minimal skew products of amenable minimal dynamical systems
• Etale groupoids arising from products of shifts of finite type
• Röver's Simple Group is of Type $F_{\infty }$
• Katsura--Exel--Pardo Groupoids and the AH~Conjecture
• Strongly continuous orbit equivalence of one-sided topological Markov shifts
• Graph and wreath products in topological full groups of full shifts
• On flow equivalence of one-sided topological Markov shifts
• Full groups of Cuntz-Krieger algebras and Higman-Thompson groups