Normal subgroups of big mapping class groups

Abstract

Let S S be a surface and let Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) be the mapping class group of S S permuting a Cantor subset K ⊂ S K \subset S . We prove two structure theorems for normal subgroups of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) . (Purity:) if S S has finite type, every normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) either contains the kernel of the forgetful map to the mapping class group of S S , or it is ‘pure’ — i.e. it fixes the Cantor set pointwise. (Inertia:) for any n n element subset Q Q of the Cantor set, there is a forgetful map from the pure subgroup PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) to the mapping class group of ( S , Q ) (S,Q) fixing Q Q pointwise. If N N is a normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) contained in PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) , its image N Q N_Q is likewise normal. We characterize exactly which finite-type normal subgroups N Q N_Q arise this way. Several applications and numerous examples are also given.