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From Fourier Analysis and Number Theory to Radon Transforms and Geometry pp 375–397Cite as

Representation Varieties of Fuchsian Groups

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Part of the book series: Developments in Mathematics ((DEVM,volume 28))

Abstract

We estimate the dimension of varieties of the form Hom(Γ, G) where Γ is a Fuchsian group and G is a simple real algebraic group, answering along the way a question of I. Dolgachev.

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References

  1. Rapinchuk, A. S.; Benyash-Krivetz, V. V.; Chernousov, V. I.: Representation varieties of the fundamental groups of compact orientable surfaces. Israel J. Math. 93 (1996), 29–71.

    Google Scholar 

  2. Dynkin, E. B.: Some properties of the system of weights of a linear representation of a semisimple Lie group. (Russian) Doklady Akad. Nauk SSSR (N.S.) 71 (1950), 221–224.

    Google Scholar 

  3. Dynkin, E. B.: Semisimple subalgebras of semisimple Lie algebras. Amer. Math. Soc. Transl. (Ser. 2) 6 (1957), 111–245.

    Google Scholar 

  4. Dynkin, E. B.: Maximal subgroups of the classical groups. Amer. Math. Soc. Transl. (Ser. 2) 6 (1957), 245–378.

    Google Scholar 

  5. Fulton, William; Harris, Joe: Representation theory. A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991.

    Google Scholar 

  6. Helgason, Sigurdur: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, Orlando, Florida, 1978.

    MATH  Google Scholar 

  7. Humphreys, James E.: Linear algebraic groups. Graduate Texts in Mathematics, No. 21. Springer-Verlag, New York-Heidelberg, 1975.

    Google Scholar 

  8. Kostant, Bertram: The principal 3-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math. 81 (1959), 973–1032.

    Google Scholar 

  9. Landazuri, Vicente; Seitz, Gary M.: On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra 32 (1974), 418–443.

    Google Scholar 

  10. Larsen, Michael: Rigidity in the invariant theory of compact groups, arXiv: math.RT/0212193.

    Google Scholar 

  11. Larsen, Michael; Lubotzky, Alex; Marion, Claude: Deformation theory and finite simple quotients of triangle groups I, preprint.

    Google Scholar 

  12. Lawther, Ross: Elements of specified order in simple algebraic groups. Trans. Amer. Math. Soc. 357 (2005), no. 1, 221–245.

    Google Scholar 

  13. Liebeck, Martin W.; Shalev, Aner: Fuchsian groups, coverings of Riemann surbfaces, subgroup growth, random quotients and random walks. J. Algebra 276 (2004), no. 2, 552–601.

    Google Scholar 

  14. Liebeck, Martin; Shalev, Aner: Fuchsian groups, finite simple groups and representation varieties. Invent. Math. 159 (2005), no. 2, 317–367.

    Google Scholar 

  15. Richardson, R. W., Jr.: A rigidity theorem for subalgebras of Lie and associative algebras. Illinois J. Math. 11 (1967) 92–110.

    Google Scholar 

  16. de Siebenthal, Jean: Sur certaines sous-groupes de rang un des groupes de Lie clos. C. R. Acad. Sci. Paris 230 (1950), 910–912.

    Google Scholar 

  17. Weil, André: Remarks on the cohomology of groups. Annals of Math. 80 (1964), 149–157.

    Google Scholar 

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Acknowledgments

ML was partially supported by the National Science Foundation and the United States-Israel Binational Science Foundation. AL was partially supported by the European Research Council and the Israel Science Foundation.

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Michael Larsen

  2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem, 91904, Israel

    Alexander Lubotzky

Authors
  1. Michael Larsen
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  2. Alexander Lubotzky
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Corresponding author

Correspondence to Alexander Lubotzky .

Editor information

Editors and Affiliations

  1. , Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Hershel M. Farkas

  2. , Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, 08544, New Jersey, USA

    Robert C. Gunning

  3. , Department of Mathematics, Temple University, Wachman Hall 608, Philadelphia, 19122, Pennsylvania, USA

    Marvin I. Knopp

  4. , Department of Mathematics, University of Michigan, Church St. 530, Ann Arbor, 48109, Michigan, USA

    B. A. Taylor

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Dedicated to the memory of Leon Ehrenpreis.

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Larsen, M., Lubotzky, A. (2013). Representation Varieties of Fuchsian Groups. In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4075-8_19

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