The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

Abstract

Let N be a connected nonorientable surface with or without boundary and punctures, and j : S → N the orientation double covering. Birman-Chillingworth, Szepietowski, and Gonçalves-Guaschi-Maldonado proved that the orientation double covering j induces an injective homomorphism ι : Mod(N ) ֒→ Mod(S) with one exception. In this paper we prove that this injective homomorphism ι is a quasi-isometric embedding as an application of the semihyperbolicity of Mod(S), which is established by Durham-Minsky-Sisto and Haettel-Hoda-Petyt. We also prove that the embedding Mod(F ′ ) ֒→ Mod(F ) induced by an inclusion of a pair of possibly nonorientable surfaces F ′ ⊂ F , well-studied by Paris-Rolfsen and Stukow, is a quasiisometric embedding.