Asymptotic mapping class groups of closed surfaces punctured along Cantor sets

Louis Funar (1) , Javier Aramayona (2)

1 IF - Institut Fourier
2 UAM - Universidad Autónoma de Madrid

We introduce subgroups B-g < H-g of the mapping class group Mod(Sigma(g)) of a closed surface of genus g >= 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 B-g and H-g are finitely presented, and that H-g is dense in Mod(Sigma(g)). We then exploit the relation with Thompson's groups to study properties B-g and 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 B-g and 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.

Domaines: Topologie géométrique [math.GT]

Louis Funar, Javier Aramayona. Asymptotic mapping class groups of closed surfaces punctured along Cantor sets. Moscow Mathematical Journal, 2021, 21 (1), pp.1--29. ⟨10.17323/1609-4514-2021-21-1-1-29⟩. ⟨hal-02330068⟩