Injective homomorphisms of mapping class groups of non-orientable surfaces

TL;DR

This paper proves that for certain non-orientable surfaces, any injective homomorphism between finite index subgroups of their mapping class groups is an inner automorphism, establishing the automorphism group's structure.

Contribution

It demonstrates that all injective homomorphisms of finite index subgroups of the mapping class group are conjugations, revealing the automorphism group structure for these surfaces.

Findings

Injective homomorphisms are conjugations by elements of the mapping class group.

The automorphism group of the mapping class group coincides with the group itself.

The abstract commensurator of the mapping class group is the group itself.

Abstract

Let $N$ be a compact, connected, non-orientable surface of genus $\rho$ with $n$ boundary components, with $\rho \ge 5$ and $n \ge 0$, and let $\mathcal{M} (N)$ be the mapping class group of $N$. We show that, if $\mathcal{G}$ is a finite index subgroup of $\mathcal{M} (N)$ and $\varphi: \mathcal{G} \to \mathcal{M} (N)$ is an injective homomorphism, then there exists $f_0 \in \mathcal{M} (N)$ such that $\varphi (g) = f_0 g f_0^{-1}$ for all $g \in \mathcal{G}$. We deduce that the abstract commensurator of $\mathcal{M} (N)$ coincides with $\mathcal{M} (N)$.