Geometric normal subgroups in mapping class groups of punctured surfaces

TL;DR

This paper demonstrates that many normal subgroups of the extended mapping class group of punctured surfaces are geometric, with their automorphism and commensurator groups isomorphic to the entire mapping class group, extending previous results.

Contribution

It establishes conditions under which normal subgroups are geometric in punctured surfaces and proves automorphism groups of associated complexes are isomorphic to the mapping class group.

Findings

Many normal subgroups are geometric with automorphism groups isomorphic to the mapping class group.

Automorphism groups of complexes associated to punctured surfaces are isomorphic to the extended mapping class group.

The results extend previous work from surfaces without punctures to punctured surfaces.

Abstract

We prove that many normal subgroups of the extended mapping class group of a surface with punctures are geometric, that is, that their automorphism groups and abstract commensurator groups are isomorphic to the extended mapping class group. In order to apply our theorem to a normal subgroup we require that the "minimal supports" of its elements satisfy a certain complexity condition that is easy to check in practice. The key ingredient is proving that the automorphism groups of many simplicial complexes associated to punctured surfaces are isomorphic to the extended mapping class group. This resolves many cases of a metaconjecture of N. V. Ivanov and extends work of Brendle-Margalit, who prove the result for surfaces without punctures.