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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

Online
ISSN
1435-5345
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Volume 2015, Issue 705

# Topological full groups of one-sided shifts of finite type

Hiroki Matui
Published Online: 2013-07-23 | DOI: https://doi.org/10.1515/crelle-2013-0041

## Abstract

We explore the topological full group $〚G〛$ of an essentially principal étale 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 étale groupoid G. Furthermore, when G is either almost finite or purely infinite, the commutator subgroup $D\left(〚G〛\right)$ 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, $〚G〛$ 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.

Revised: 2013-05-06

Published Online: 2013-07-23

Published in Print: 2015-08-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2015, Issue 705, Pages 35–84, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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