#### Volume 26, issue 3 (2022)

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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
##### 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 ${T}^{♯},{T}^{\ast }$ introduced by Funar and Kapoudjian, and the braided Houghton groups $\mathrm{br}{H}_{n}$ 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 ${T}^{\ast }$ are of type ${F}_{\infty }$ and that $\mathrm{br}{H}_{n}$ 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
Primary: 20F65
Secondary: 20J05