Homological stability for the ribbon Higman--Thompson groups

TL;DR

This paper establishes homological stability for the ribbon Higman--Thompson groups by generalizing asymptotic mapping class groups and connecting them to big mapping class groups, extending previous stability results to a surface setting.

Contribution

It introduces a new model linking asymptotic mapping class groups to ribbon Higman--Thompson groups and proves their homological stability, extending prior work to dense subgroups of big mapping class groups.

Findings

Proved homological stability for ribbon Higman--Thompson groups.

Connected asymptotic mapping class groups with big mapping class groups.

Extended Szymik--Wahl's stability results to surface-related groups.

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.