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

Let N be a connected nonorientable surface with or without boundary and punctures, and j S→ N be the orientation double covering. It has previously been proved that j induces an embedding ι(N)(S) with one exception. In this paper, we prove that the injective homomorphism ι is a quasi-isometric embedding. The proof is based on the semihyperbolicity of Mod(S) , which has already been established. We also prove that the embedding Mod(F') Mod(F) induced by an inclusion of a pair of possibly nonorientable surfaces F' ⊂ F is a quasi-isometric embedding.