Asymptotically rigid mapping class groups, I : Finiteness properties of braided Thompson’s and Houghton’s groups

Genevois, Anthony; Lonjou, Anne; Urech, Christian

doi:10.2140/gt.2022.26.1385

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Asymptotically rigid mapping class groups, I: Finiteness properties of braided Thompson's and Houghton's groups

Anthony Genevois, Anne Lonjou and Christian Urech

Geometry & Topology 26 (2022) 1385–1434

DOI: 10.2140/gt.2022.26.1385

Abstract

We study the asymptotically rigid mapping class groups of infinitely punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy–Thompson groups $T^{♯}, T^{*}$ introduced by Funar and Kapoudjian, and the braided Houghton groups $br H_{n}$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube stabilizers isomorphic to finite extensions of braid groups. As an application, we prove conjectures of Funar–Kapoudjian and Degenhardt by showing that $T^{♯}$ and $T^{*}$ are of type $F_{\infty}$ and that $br H_{n}$ is of type $F_{n - 1}$ but not of type $F_{n}$ .

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Keywords

Thompson groups, Houghton groups, braid groups, big mapping class groups, asymptotically rigid mapping class groups, cube complexes

Mathematical Subject Classification

Primary: 20F65

Secondary: 20J05

References

Publication

Received: 28 October 2020

Revised: 21 December 2020

Accepted: 21 January 2021

Published: 3 August 2022

Proposed: Mladen Bestvina

Seconded: David M Fisher, Anna Wienhard

Authors

Anthony Genevois
Institut Montpelliérain Alexander Grothendieck Université de Montpellier Montpellier France

Anne Lonjou
Département de Mathématiques Faculté des Sciences d’Orsay Université Paris-Saclay Orsay France

Christian Urech
Institut de Mathématiques École Polytechniqe Fédérale de Lausanne Lausanne Switzerland

Volume 26, issue 3 (2022)