Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersen, J.E.: Fixed points of the mapping class group in the \({\rm SU}(n)\) moduli spaces. Proc. Am. Math. Soc. 125(5), 1511–1515 (1997)
Bigelow, S.J.: Braid groups are linear. J. Am. Math. Soc 14(2), 471–486 (2001). (electronic)
Bigelow, S.J., Budney, R.D.: The mapping class group of a genus two surface is linear. Algebr. Geom. Topol. 1, 699–708 (2001). (electronic)
Birman, J.S.: Braids, links, and mapping class groups. Princeton University Press, Princeton (1975)
Brendle, T.E., Hamidi-Tehrani, H.: On the linearity problem for mapping class groups. Algebr. Geom. Topol. 1, 445–468 (2001)
Farb, B., Lubotzky, A., Minsky, Y.: Rank-1 phenomena for mapping class groups. Duke Math. J. 106(3), 581–597 (2001)
Farb, B., Margalit, D.: A primer on mapping class groups, Princeton mathematical series, vol. 49. Princeton University Press, Princeton (2012)
Formanek, E., Procesi, C.: The automorphism group of a free group is not linear. J. Algebr. 149(2), 494–499 (1992)
Franks, J., Handel, M.: Triviality of some representations of MCG\((S_g)\) in \(GL(n, {\mathbb{C}})\), Diff\((S^2)\) and Homeo\(({\mathbb{T}}^2)\). Proc. Am. Math. Soc. 141(9), 2951–2962 (2013)
Gervais, S.: A finite presentation of the mapping class group of a punctured surface. Topology 40(4), 703–725 (2001)
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)
Korkmaz, M.: On the linearity of certain mapping class groups. Turkish J. Math. 24(4), 367–371 (2000)
Korkmaz, M.: Low-dimensional linear representations of mapping class groups, preprint. arXiv:1104.4816 (2011)
Korkmaz, M.: The symplectic representation of the mapping class group is unique, preprint. arXiv:1108.3241 (2011)
Krammer, D.: The braid group \(B_4\) is linear. Invent. Math. 142(3), 451–486 (2000)
Krammer, D.: Braid groups are linear. Ann. Math. (2) 155(1), 131–156 (2002)
Paris, L., Rolfsen, D.: Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521, 47–83 (2000)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kasahara, Y. On visualization of the linearity problem for mapping class groups of surfaces. Geom Dedicata 176, 295–304 (2015). https://doi.org/10.1007/s10711-014-9968-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-014-9968-0