Automorphisms of the mapping class group of a nonorientable surface

TL;DR

This paper proves that for nonorientable surfaces of genus at least 5 with equal complexity, all isomorphisms between their mapping class groups are geometric, induced by surface diffeomorphisms, extending Ivanov's theorem.

Contribution

It establishes that isomorphisms of mapping class groups of nonorientable surfaces of the same complexity are always geometric, generalizing and strengthening previous results.

Findings

Every isomorphism between such groups is induced by a surface diffeomorphism.

The result extends Ivanov's theorem to nonorientable surfaces.

It improves upon the authors' previous work on this topic.

Abstract

Let $S$ be a nonorientable surface of genus $g\ge 5$ with $n\ge 0$ punctures, and $\Mcg(S)$ its mapping class group. We define the complexity of $S$ to be the maximum rank of a free abelian subgroup of $\Mcg(S)$. Suppose that $S_1$ and $S_2$ are two such surfaces of the same complexity. We prove that every isomorphism $\Mcg(S_1)\to\Mcg(S_2)$ is induced by a diffeomorphism $S_1\to S_2$. This is an analogue of Ivanov's theorem on automorphisms of the mapping class groups of an orientable surface, and also an extension and improvement of the first author's previous result.