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 $\varphi : \mathcal G\rightarrow \mathcal M(N)$ 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 $\mathcal M(N)$.