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.

Big mapping class groups with uncountable integral homology

“…When g D 0, a positive answer follows from [24,Theorem B]. The answer in degree one (and for any g) has been proven to be positive in [8,Theorem 2.3]. On the other hand, the answer would be negative if we considered the sphere instead of S g;1 , since H 2 .Map.S 2 X C // Š Z=2 by [7,Theorem A.2].…”

“…In particular, in all of these examples, the homology groups H i .Map.S // are finitely generated for each i . Some earlier results on the homology of big mapping class groups -in degrees 1 and 2 -include: H 1 .Map.S X C// Š H 1 .Map.S // if C is a Cantor set embedded in the interior of a finite-type surface S [8] (see also [28] for three special cases of this) and H 2 .Map.S 2 X C // Š Z=2 [7].…”

Stable commutator length on big mapping class groups

“…We also note that the surfaces covered in Theorem 1.4 include finite‐type surfaces with a Cantor set of points removed. The abelianization of the mapping class group of these surfaces has been previously studied and is known by [12].…”