On visualization of the linearity problem for mapping class groups of surfaces
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We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear representations of the fundamental group of the corresponding closed surface. For the closed case, the condition is described in terms of the vector space generated by the isotopy classes of essential simple closed curves on the corresponding surface. The latter condition also describes the linearity for the mapping class group of compact orientable surface with boundary, up to center.
KeywordsMapping class group Linear representation Deformation space
Mathematical Subject Classification57M50 20F65 57M07
The author is grateful to Louis Funar and Makoto Sakuma for valuable discussions and comments. He is grateful to Masatoshi Sato for an enlightening conversation. He is grateful to the referee for helpful comments. The author was partially supported by the Grant-in-Aid for Scientific Research (C) (No.23540102) from the Japan Society for Promotion of Sciences.
- 7.Farb, B., Margalit, D.: A primer on mapping class groups, Princeton mathematical series, vol. 49. Princeton University Press, Princeton (2012)Google Scholar
- 11.Goldman, W.M.: Mapping class group dynamics on surface group representations, Problems on mapping class groups and related topics. In: Proceedings of the Symposium in Pure Mathematics, vol. 74, pp. 189–214. American Mathematical Society (2006)Google Scholar
- 13.Korkmaz, M.: Low-dimensional linear representations of mapping class groups, preprint. arXiv:1104.4816 (2011)
- 14.Korkmaz, M.: The symplectic representation of the mapping class group is unique, preprint. arXiv:1108.3241 (2011)