Isomorphisms between big mapping class groups

TL;DR

This paper proves that isomorphisms between big mapping class groups of infinite-type surfaces are realized by surface homeomorphisms, extending to automorphisms of finite-index subgroups and automorphisms of curve complexes.

Contribution

It establishes that all automorphisms of big mapping class groups and their curve complexes are induced by homeomorphisms, revealing their rigid structure.

Findings

Isomorphisms between big mapping class groups are induced by surface homeomorphisms.

Finite-index subgroups have finite outer automorphism groups.

Automorphisms of curve complexes are induced by homeomorphisms.

Abstract

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.