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 ⊂ 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 NQ is likewise normal. We characterize exactly which finite-type normal subgroups NQ arise this way.Several applications and numerous examples are also given. Contents 1. Introduction 1 1.1. Statement of results 2 Acknowledgment 3 2. The Purity Theorem 3 2.1. Applications 4 2.2. Proof of the Purity Theorem when S has at most one puncture 5 2.3. Proof of the Purity Theorem in general 11 3. The Inertia Theorem 12 3.1. Inert subgroups 13 3.2. Proof of the Inertia Theorem 15 3.3. Operations on normal subgroups 19 References 20

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].…”