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 $Mod({\mathrm{\Sigma }}_g)$ 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 $Mod({\mathrm{\Sigma }}_g)$. 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.