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 $j\colon S\rightarrow N$ be the orientation double covering. It has previously been proved that the orientation double covering $j$ induces an embedding $ι\colon\mathrm{Mod}(N)$ $\hookrightarrow$ $\mathrm{Mod}(S)$ with one exception. In this paper, we prove that this injective homomorphism $ι$ is a quasi-isometric embedding. The proof is based on the semihyperbolicity of $\mathrm{Mod}(S)$, which has already been established. We also prove that the embedding $\mathrm{Mod}(F') \hookrightarrow \mathrm{Mod}(F)$ induced by an inclusion of a pair of possibly nonorientable surfaces $F' \subset F$ is a quasi-isometric embedding.

Figures (2)

• Figure 1: We represent $S_{g-1,2p}^{2b}$ in $\mathbb{R}^{3}$ as the left (resp. right) surface when $g-1$ is odd (resp. even), where the $2b$ circles are the boundary components and the $2p$ points are the punctures of $S_{g-1, 2p}^{2b}$.
• Figure 2: The orientation double covering $j\colon S_{g-1, 2p}^{2b}\rightarrow N_{g, p}^{b}$ when $g-1$ is odd (left) and even (right).

Theorems & Definitions (25)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• proof : Proof of Lemma \ref{['cent_qconvex']}
• Lemma 2.6
• Remark 2.7
• Lemma 2.8