Injective homomorphisms of mapping class groups of non-orientable surfaces

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 $\phi :\mathcal{G}\to \mathcal{M}(N)$ is an injective homomorphism, then there exists $f_0\in \mathcal{M}(N)$ such that $\phi (g)=f_{0}g{f}_0^{-1}$ for all $g\in \mathcal{G}$. We deduce that the abstract commensurator of $\mathcal{M}(N)$ coincides with $\mathcal{M}(N)$.