Normal subgroups of big mapping class groups
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.
Cite
@article{arxiv.2110.07839,
title = {Normal subgroups of big mapping class groups},
author = {Danny Calegari and Lvzhou Chen},
journal= {arXiv preprint arXiv:2110.07839},
year = {2022}
}
Comments
v3: revised according to referee's suggestions, final version to appear in TAMS. 21 pages, 3 figures