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Geometriae Dedicata

, Volume 176, Issue 1, pp 295–304 | Cite as

On visualization of the linearity problem for mapping class groups of surfaces

  • Yasushi KasaharaEmail author
Original Paper

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.

Keywords

Mapping class group Linear representation Deformation space 

Mathematical Subject Classification

57M50 20F65 57M07 

Notes

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.

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsKochi University of TechnologyTosayamada, Kami CityJapan

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