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

TL;DR

This paper investigates the finiteness properties of certain asymptotically rigid mapping class groups related to infinitely-punctured surfaces, providing new constructions and confirming conjectures about their algebraic types.

Contribution

It introduces an elementary construction of a contractible cube complex for these groups and proves their finiteness properties, confirming existing conjectures.

Findings

Groups $T^lat,T^ atural$ are of type $F_$

Groups $ ext{br}H_n$ are of type $F_{n-1}$ but not $F_n$

Constructed a contractible cube complex with specific stabilizers

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 $T^\sharp,T^\ast$ 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 $T^\sharp,T^\ast$ are of type $F_\infty$ and that $\mathrm{br}H_n$ is of type $F_{n-1}$ but not of type $F_n$.