The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover
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| Format: | Preprint |
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2021
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| _version_ | 1866916228427153408 |
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| author | Katayama, Takuya Kuno, Erika |
| author_facet | Katayama, Takuya Kuno, Erika |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_11839 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover Katayama, Takuya Kuno, Erika Geometric Topology Group Theory 20F65, 20F67, 57K20 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. |
| title | The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover |
| topic | Geometric Topology Group Theory 20F65, 20F67, 57K20 |
| url | https://arxiv.org/abs/2101.11839 |