Topological full groups of one-sided shifts of finite type

2012/10/22 by Hiroki Matui, Matui, Hiroki · 8 citations
Mathematics · #37B10 #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1210.5800

openalex publication_date 2012/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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_∞, 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.

Citations

• Cartan subalgebras in C*-algebras
• On Groups of Measure Preserving Transformations. I
• Some remarks on topological full groups of Cantor minimal systems II
• Full groups of one-sided topological Markov shifts

Cited by

• Limits in categories of étale groupoids and pseudogroups
• Boundary actions of outer automorphism groups of Thompson-like groups
• Ample groupoids that are neither almost finite nor purely infinite
• Non-strong ergodicity of canonical actions of the Thompson groups
• Chern character for torsion-free ample groupoids
• Topological Full Groups of Irreducible Edge Shifts have Solvable Conjugacy Problem
• On Similarity Structure Groups and their W^* and C^*-Algebras
• Cocompact unfolding trees

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• Etale groupoids arising from products of shifts of finite type
• Topological full groups of etale groupoids
• Rigidity properties of full groups of pseudogroups over the Cantor set
• Topological full groups and t.d.l.c. completions of Thompson's V
• Left regular representations of Garside categories II. Finiteness properties of topological full groups