Homological stability for the ribbon Higman--Thompson groups
Rachel Skipper, Xiaolei Wu
TL;DR
The paper generalizes asymptotic mapping class groups to surject onto the ribbon Higman–Thompson groups and proves homological stability for these dense subgroups of big mapping class groups, extending Szymik–Wahl’s results to a surface setting. Using a geometric model built from d-rigid structures and asymptotic mapping class groups, the authors identify BV_{d,r}(D) with RV^+_{d,r} and HV_{d,r}(D) with RV_{d,r}, establishing density in Map(Σ^{∞}_{d,r}). They develop a homogeneous category framework and prove high connectivity for the associated complex W_r(D,D^{∞}_{d,1}), applying the RWW17 stability theorem to obtain isomorphisms H_i(RV^+_{d,r},M) ≅ H_i(RV^+_{d,r+1},M) for all i and large r (with an improved range depending on d). This yields the first homological stability result for dense subgroups of big mapping class groups and generalizes prior Higman–Thompson stability to the ribbon and surface contexts. The work also lays groundwork for computing ribbon–Higman–Thompson homology and informs related acyclicity questions in braided settings.
Abstract
We generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman--Thompson groups, answering a question of Aramayona and Vlamis in the case of the Higman--Thompson groups. When the underlying surface is a disk, these new asymptotic mapping class groups can be identified with the ribbon and oriented ribbon Higman--Thompson groups. We use this model to prove that the ribbon Higman--Thompson groups satisfy homological stability, providing the first homological stability result for dense subgroups of big mapping class groups. Our result can also be treated as an extension of Szymik--Wahl's work on homological stability for the Higman--Thompson groups to the surface setting.