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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

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 brHn 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 and that brHn is of type Fn1 but not of type Fn.

Thompson groups, Houghton groups, braid groups, big mapping class groups, asymptotically rigid mapping class groups, cube complexes
Mathematical Subject Classification
Primary: 20F65
Secondary: 20J05
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
Anthony Genevois
Institut Montpelliérain Alexander Grothendieck
Université de Montpellier
Anne Lonjou
Département de Mathématiques
Faculté des Sciences d’Orsay
Université Paris-Saclay
Christian Urech
Institut de Mathématiques
École Polytechniqe Fédérale de Lausanne