2013/07/22 by Hiroki Matui · 2 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.1515/crelle-2013-0041
openalex publication_date 2013/07/22 · crossref created 2013/07/22 · crossref issued 2013/07/23 · crossref published 2013/07/23 · crossref published-online 2013/07/23 · crossref published-print 2015/08/01 · crossref deposited 2023/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21 · crossref indexed 2026/07/31
Abstract We explore the topological full group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>〚</m:mo> <m:mi>G</m:mi> <m:mo>〛</m:mo> </m:mrow> </m:math> \llbracket G\rrbracket of an essentially principal étale groupoid G on a Cantor set. When G is minimal, we show that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>〚</m:mo> <m:mi>G</m:mi> <m:mo>〛</m:mo> </m:mrow> </m:math> \llbracket G\rrbracket (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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>D</m:mi> <m:mo>(</m:mo> <m:mo>〚</m:mo> <m:mi>G</m:mi> <m:mo>〛</m:mo> <m:mo>)</m:mo> </m:mrow> </m:math> D(\llbracket G\rrbracket ) 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 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>〚</m:mo> <m:mi>G</m:mi> <m:mo>〛</m:mo> </m:mrow> </m:math> \llbracket G\rrbracket is thought of as a generalization of the Higman–Thompson group. We prove that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>〚</m:mo> <m:mi>G</m:mi> <m:mo>〛</m:mo> </m:mrow> </m:math> \llbracket G\rrbracket 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>〚</m:mo> <m:mi>G</m:mi> <m:mo>〛</m:mo> </m:mrow> </m:math> \llbracket G\rrbracket is calculated and described in terms of the homology groups of G .