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

摘要

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 $T^sharp,T^ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $mathop{rm br} H_n$ 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 $T^sharp$ and $T^ast$ are of type $F_infty$ and that $mathop{rm br} H_n$ is of type $F_{n-1}$ but not of type $F_n$.