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Cohomological finiteness conditions and centralisers in generalisations of Thompson’s group V

Conchita Martínez-Pérez, Francesco Matucci and Brita E. A. Nucinkis
From the journal Forum Mathematicum

Abstract

We consider generalisations of Thompson’s group V, denoted by Vr(Σ), which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, Vr(Σ) is the full automorphism group of a Cantor algebra. Under some further minor restrictions, we prove that these groups are of type F and that this implies that also centralisers of finite subgroups are of type F.

MSC 2010: 20J05

Communicated by Frederick R. Cohen


Funding statement: This work was partially funded by an LMS Scheme 4 Grant 41209. The first named author was supported by Gobierno de Aragón, European Regional Development Funds and MTM2010-19938-C03-03. The second author gratefully acknowledges the Fondation Mathématique Jacques Hadamard (ANR-10-CAMP-0151-02 – FMJH – Investissement d’Avenir) for the support received during the development of this work.

We would like to thank Dessislava Kochloukova for helpful discussions regarding Section 3, Claas Röver for getting us to think about Theorem 2.5 and the anonymous referee for a very carefully reading an earlier version of this paper. We are also indebted to this referee for pointing out Lemma 4.8 to us.

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Received: 2014-10-6
Revised: 2015-2-3
Published Online: 2015-10-20
Published in Print: 2016-9-1

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