standard ternary Cantor set is in fact isomorphic to its smooth mapping class group [FN18, Corollary 2]. In [AV20, Question 5.37], the following question was raised by Aramayona and Vlamis.
Question. Are there other geometrically defined subgroups of $\text{Map}(\Sigma_g)$ which surject to other interesting classes of subgroups of homeomorphism group of the Cantor set, such as the Higman-Thompson groups, Neretin groups, etc?
We proceed to construct two new classes of asymptotic mapping class groups, one of which answers their question in the case of Higman-Thompson groups while the other family surjects to the symmetric Higman-Thompson groups $V_{d,r}(\mathbb{Z}/2\mathbb{Z})$.
Theorem (3.17, 3.20). Let $\Sigma$ be any compact surface and $C$ be a Cantor set which lies in the interior of a disk in $\Sigma$. Then the mapping class group $\text{Map}(\Sigma \setminus C)$ contains the following two families of dense subgroups: the asymptotic mapping class groups $\mathcal{B}V_{d,r}(\Sigma)$, which surject to the Higman-Thompson groups $V_{d,r}$, and the half-twist asymptotic mapping class groups $\mathcal{H}V_{d,r}(\Sigma)$, which surject to the symmetric Higman-Thompson groups $V_{d,r}(\mathbb{Z}/2\mathbb{Z})$.
When $\Sigma$ is the disk, we identify $\mathcal{H}V_{d,r}(\Sigma)$ with the ribbon Higman-Thompson group $RV_{d,r}$ and $\mathcal{B}V_{d,r}(\Sigma)$ with the oriented ribbon Higman-Thompson group $RV_{d,r}^+$ (cf. Theorem 3.24). Using this geometric model for the ribbon Higman-Thompson groups, we are able to prove the following.
Theorem (4.30, 4.31). Suppose $d \ge 2$. Then the inclusion maps induce isomorphisms
in homology in all dimensions $i \ge 0$, for all $r \ge 1$ and for all $H_1(RV_{d,\infty})$-modules $M$. The same also holds for the oriented ribbon Higman-Thompson groups $RV_{d,r}^+$.
To the best of our knowledge, this is the first homological stability result for dense subgroups of big mapping class groups. Our proof uses a recent convenient framework given by Randal-Williams and Wahl [RWW17]. The core of the proof is similar to [SW19], but with new technical difficulties arising from infinite type surface topology. We hope our result here can be further used to calculate the homology of ribbon Higman-Thompson groups and shed light on the question whether braided $V$ is acyclic.
Outline of paper. In Section 1, we describe the connectivity tools that will be necessary for the remainder of the paper. In Section 2, we introduce the definition of the Higman-Thompson, ribbon Higman-Thompson, and oriented ribbon Higman-Thompson groups using paired forest diagrams to define the elements. In Section 3, we generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman-Thompson groups. And finally, in Section 4, we prove homological stability for the ribbon Higman-Thompson groups and their oriented version.
Notation and convention. All surfaces in this paper are assumed to be connected and orientable unless otherwise stated. Given a simplicial complex $X$ and a cell $\sigma \in X$, we denote the link of $\sigma$ in $X$ by $Lk_X(\sigma)$ (resp. the star of $\sigma$ by $\text{St}_X(\sigma)$). When the situation is clear, we quite often omit $X$ and simply denote the link by $Lk(\sigma)$ and the star by $\text{St}(\sigma)$. We also use the convention that $(-1)$-connected means non-empty and that every space is $(-2)$-connected. In particular, the empty set is $(-2)$-connected. Finally, we adopt the convention that elements in groups are multiplied from left to right.
Acknowledgements. The first part of this project was done while the first author was a visitor in the Unité de mathématiques pures et appliquées at the ENS de Lyon and during a visit to the University of Bonn. She thanks them for their hospitality. She was also supported by the GIF, grant I-198-304.1-2015, “Geometric exponents of random walks and intermediate growth groups” and NSF DMS-2005297 “Group Actions on Trees and Boundaries of Trees”.