Injective homomorphisms of mapping class groups of non-orientable surfaces

Elmas Irmak (1) , Luis Paris (2)

1 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 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).

Domaines: Topologie géométrique [math.GT]; Théorie des groupes [math.GR]

Elmas Irmak, Luis Paris. Injective homomorphisms of mapping class groups of non-orientable surfaces. 2017. ⟨hal-01570874⟩