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

Abstract

This article is dedicated to the study of 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-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjian's and Degenhardt's conjectures by showing that $T^{♯},T^{\ast }$ are of type $F_{\infty }$ and that $\mathrm{br}{H}_n$ is of type $F_{n-1}$ but not of type $F_n$.

arXiv comment36 pages, 11 figures. Comments are welcome!

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• Groups of piecewise projective homeomorphisms
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• The braided Ptolemy-Thompson group is finitely presented
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Cited by in corpus (6)

• Braided Thompson groups with and without quasimorphisms
• Braided Brin-Thompson Groups
• Braiding groups of automorphisms and almost-automorphisms of trees
• Homological stability for the ribbon Higman--Thompson groups
• Finiteness properties for relatives of braided Higman--Thompson groups
• Asymptotically rigid mapping class groups II: strand diagrams and nonpositive curvature