Abstract
Let N be a closed nonorientable surface with or without marked points. In this paper we prove that, for every finite full subgraph \(\Gamma \) of \({\mathcal {C}}^{\textrm{two}}(N)\), the right-angled Artin group on \(\Gamma \) can be embedded in the map** class group of N. Here, \({\mathcal {C}}^{\textrm{two}}(N)\) is the subgraph, induced by essential two-sided simple closed curves in N, of the ordinary curve graph \({\mathcal {C}}(N)\). In addition, we show that there exists a finite graph \(\Gamma \) which is not a full subgraph of \({\mathcal {C}}^{\textrm{two}}(N)\) for some N, but the right-angled Artin group on \(\Gamma \) can be embedded in the map** class group of N.
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References
Atalan, F., Korkmaz, M.: Automorphisms of curve complexes on nonorientable surfaces. Groups Geom. Dyn. 8(1), 39–68 (2014)
Baik, H., Kim, S., Koberda, T.: Unsmoothable group actions to one-manifolds. J. Eur. Mat. Soc. 21(8), 2333–2353 (2019)
Bestvina, M., Fujiwara, K.: Quasi-homomorphisms on map** class groups. Glas. Mat. Ser. III 42(62)(1), 213–236 (2007)
Crisp, J., Wiest, B.: Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups. Algebr. Geom. Topol. 4, 43–472 (2004)
Davis, M.: Geometry and Topology of Coxeter Groups. Princeton University Press, New Jersey (2008)
Gonçalves, D.L., Guaschi, J., Maldonado, M.: Embeddings and the (virtual) cohomological dimension of the braid and map** class groups of surfaces. Conflu. Math. 10(1), 41–61 (2018)
Kapovich, M.: RAAGs in Ham. Geom. Funct. Anal. 22, 733–755 (2012)
Katayama, T., Kuno, E.: The RAAGs on the complement graphs of linear forests in map** class groups, preprint. ar** class groups. Geom. Funct. Anal. 22(6), 1541–1590 (2012). https://doi.org/10.1007/s00039-012-0198-z
Korkmaz, M.: Map** class groups of nonorientable surfaces. Geom. Dedicata 89, 109–133 (2002)
Kim, S., Koberda, T.: An obstruction to embedding right-angled Artin groups in map** class groups. Int. Mat. Res. Not. IMNR(14), 912–3918 (2014)
Kim, S., Koberda, T.: Anti-trees and right-angled Artin subgroups of braid groups. Geom. Topol. 19, 3289–3306 (2015)
Kim, S., Koberda, T.: Right-angled Artin groups and finite subgraphs of curve graphs. Osaka J. Math. 53(3), 705–716 (2016)
Kim, S., Koberda, T., Rivas, C.: Direct products, overlap** actions, and critical regularity. J. Mod. Dyn. 17, 285–304 (2021)
Kuno, E.: Abelian subgroups of the map** class groups for non-orientable surfaces. Osaka J. Math. 56(1), 91–100 (2019)
Masur, H., Schleimer, S.: The geometry of the disk complex. J. Amer. Math. Soc. 26(1), 1–62 (2013). https://doi.org/10.1090/S0894-0347-2012-00742-5
Niblo, G., Wise, D.: Subgroup separability, knot groups and graph manifolds. Proc. Amer. Math. Soc. 129(3), 685–693 (2000)
Runnels, I.: Effective generation of right-angled Artin groups in map** class groups. Geom. Dedicata 214, 277–294 (2021)
Seo, D.: Powers of Dehn twists generating right-angled Artin groups. Algebr. Geom. Topol. 21, 1511–1533 (2021)
Stukow, M.: Subgroups generated by two Dehn twists on a nonorientable surface. Topol. Proc. 50, 151–201 (2017)
Thurston, W.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. N.S 19(2), 417–431 (1988)
Wu, Y.: Canonical reducing curves of surface homeomorphism. Acta Math. Sin. N.S 3(4), 305–313 (1987)
Acknowledgements
The authors are grateful to Anthony Genevois, Sang-hyun Kim and Donggyun Seo for helpful comments on Sect. 3 and the appendix. The authors also thank to Naoto Shida for drawing figures. The first author was supported by JSPS KAKENHI, the grant number 20J01431, and the second author was supported by JST, ACT-X, the grant number JPMJAX200D, Japan, and partially supported by JSPS KAKENHI Grant-in-Aid for Early-Career Scientists, Grant Number 21K13791.
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Katayama, T., Kuno, E. Right-angled Artin groups and curve graphs of nonorientable surfaces. Geom Dedicata 217, 62 (2023). https://doi.org/10.1007/s10711-023-00788-w
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DOI: https://doi.org/10.1007/s10711-023-00788-w