Injective homomorphisms of mapping class groups of non-orientable surfaces

Abstract

Let N be a compact, connected, non-orientable surface of genus ρ with n boundary components, with ρ ≥ 5 and n ≥ 0, and let M(N) be the mapping class group of N. We show that, if G is a finite index subgroup of M(N) and ϕ : G → M(N) is an injective homomorphism, then there exists f ∈ M(N) such that ϕ(g) = f gf^( -1) for all g ∈ G. We deduce that the abstract commensurator of M(N) coincides with M(N).