Résumé
Geom. Topol. 26 (2022) 1385-1434 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$F_n$ .