Normal subgroups of big mapping class groups

TL;DR

This paper characterizes the structure of normal subgroups in big mapping class groups, revealing conditions under which they contain kernels or fix Cantor sets, with applications to understanding their algebraic properties.

Contribution

It provides the first comprehensive structure theorems for normal subgroups of big mapping class groups, including the concepts of purity and inertia, and characterizes finite-type normal subgroups.

Findings

Normal subgroups either contain the kernel or are pure, fixing the Cantor set pointwise.

Existence of forgetful maps from pure subgroups to mapping class groups of finite subsets.

Characterization of finite-type normal subgroups arising from these maps.

Abstract

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