Asymptotic mapping class groups of closed surfaces punctured along Cantor sets

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.