Geodesic
The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover
Takuya Katayama1 orcid ; Erika Kuno2 orcid
1Gakushuin University, Tokyo, Japan
2Osaka University, Osaka, Japan
Groups, geometry, and dynamics, Tome 18 (2024) no. 2, pp. 407-418

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DOI

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:Mod(N)↪Mod(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.
DOI : 10.4171/ggd/776
Classification : 20F65, 20F67, 57K20
Mots-clés : mapping class group, symmetric mapping class group, nonorientable surface, semihyperbolicity, subgroup distortion

Takuya Katayama  1   ; Erika Kuno  2

1 Gakushuin University, Tokyo, Japan
2 Osaka University, Osaka, Japan 
Takuya Katayama; Erika Kuno. The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover. Groups, geometry, and dynamics, Tome 18 (2024) no. 2, pp. 407-418. doi: 10.4171/ggd/776
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     author = {Takuya Katayama and Erika Kuno},
     title = {The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover},
     journal = {Groups, geometry, and dynamics},
     pages = {407--418},
     year = {2024},
     volume = {18},
     number = {2},
     doi = {10.4171/ggd/776},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/776/}
}
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