# Moscow Mathematical Journal

Volume 21, Issue 1, January–March 2021  pp. 1–29.

Asymptotic Mapping Class Groups of Closed Surfaces Punctured along Cantor Sets

Authors:  Javier Aramayona (1) and Louis Funar (2)
Author institution:(1) Universidad Autónoma de Madrid & ICMAT, C. U. de Cantoblanco. 28049, Madrid, Spain
(2) Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France

Summary:

We introduce subgroups $\mathcal{B}_g< \mathcal{H}_g$ of the mapping class group $\mathrm{Mod}(\Sigma_g)$ of a closed surface of genus $g \ge 0$ with a Cantor set removed, which are extensions of Thompson's group $V$ by a direct limit of mapping class groups of compact surfaces of genus $g$. We first show that both $\mathcal{B}_g$ and $\mathcal{H}_g$ are finitely presented, and that $\mathcal{H}_g$ is dense in $\mathrm{Mod}(\Sigma_g)$. We then exploit the relation with Thompson's groups to study properties $\mathcal{B}_g$ and $\mathcal{H}_g$ in analogy with known facts about finite-type mapping class groups. For instance, their homology coincides with the stable homology of the mapping class group of genus $g$, every automorphism is geometric, and every homomorphism from a higher-rank lattice has finite image.

In addition, the same connection with Thompson's groups will also prove that $\mathcal{B}_g$ and $\mathcal{H}_g$ are not linear and do not have Kazhdan's Property (T), which represents a departure from the current knowledge about finite-type mapping class groups.

2010 Math. Subj. Class. 57M50, 20F65.

Keywords: Surface, Cantor set, homeomorphism.