# Asymptotic mapping class groups of closed surfaces punctured along Cantor sets

## TL;DR

This paper introduces new subgroups of the mapping class group for surfaces with Cantor set punctures, showing they are finitely presented, dense, and share properties with finite-type mapping class groups, while also lacking linearity and Property (T).

## Contribution

It defines and analyzes asymptotic mapping class groups related to Cantor set punctured surfaces, revealing their algebraic and homological properties and contrasting them with classical finite-type groups.

## Key findings

- Both groups are finitely presented.
- The groups are dense in the mapping class group.
- They are not linear and lack Kazhdan's Property (T).

## Abstract

We introduce subgroups ${\mathcal{B}}_g< {\mathcal H}_g$ of the mapping class group $Mod(\Sigma_g)$ 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 $Mod(\Sigma_g)$. 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.