Injective homomorphisms of mapping class groups of non-orientable surfaces

Elmas Irmak (1) , Luis Paris (2)

1 Department of Mathematics and Statistics [Bowling Green State University]
2 IMB - Institut de Mathématiques de Bourgogne [Dijon]

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 f0∈M(N) such that φ(g)=f0gf−10 for all g∈G. We deduce that the abstract commensurator of M(N) coincides with M(N).

Domaines: Mathématiques [math]

Elmas Irmak, Luis Paris. Injective homomorphisms of mapping class groups of non-orientable surfaces. Geometriae Dedicata, 2019, 198 (1), pp.149-170. ⟨10.1007/s10711-018-0334-5⟩. ⟨hal-02173806⟩