Geometriae Dedicata

, Volume 120, Issue 1, pp 179–191 | Cite as

The Action of the Mapping Class Group on Maximal Representations

  • Anna WienhardEmail author
Original Paper


Let Γ g be the fundamental group of a closed oriented Riemann surface Σ g , g ≥ 2, and let G be a simple Lie group of Hermitian type. The Toledo invariant defines the subset of maximal representations Repmax g , G) in the representation variety Rep(Γ g , G). Repmax g , G) is a union of connected components with similar properties as Teichmüller space \(\mathcal{T}(\Sigma_g) = {\rm Rep}_{\max}(\Gamma_g, {\rm PSL}(2,\mathbb{R}))\). We prove that the mapping class group \(Mod_{\Sigma_g}\) acts properly on Repmax g , G) when \(G= {\rm Sp}(2n,\mathbb{R})\), SU(n,n), SO*(4n), Spin(2,n).


Mapping class group Modular group Representation variety Maximal representations Toledo invariant Teichmüller space 

Mathematics Subject Classifications (2000)

Primary 20H10 Secondary 32M15 32G15 


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  1. 1.
    Bradlow S.B., Garcí a Prada O., Gothen P.B. (2003) Surface group representations in PU(p,q) and Higgs bundles. J. Differential. Geom. 64(1): 111–170MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bradlow, S.B., Garcí a Prada, O., Gothen, P.B. Homotopy groups of moduli spaces of representations. arXiv:math.AG/0506444, (Topology to appear)Google Scholar
  3. 3.
    Burger, M., Labourie, F., Iozzi, A., Wienhard, A. Maximal representations of surface groups: Symplectic Anosov structures. Pure and Appl Math Quaterly. Special Issue: In Memory of Armand Borel. 1(2), 555–601 (2005)Google Scholar
  4. 4.
    Burger, M., Iozzi, A., Wienhard, A. Surface group representations with maximal Toledo invariant. PreprintGoogle Scholar
  5. 5.
    Burger, M., Iozzi, A., Wienhard, A. Tight embeddings. PreprintGoogle Scholar
  6. 6.
    Burger M., Iozzi A., Wienhard A. (2003) Surface group representations with maximal Toledo invariant. C. R. Acad. Sci. Paris, Sér. I 336, 387–390MathSciNetzbMATHGoogle Scholar
  7. 7.
    Clerc J.L., Ørsted B. (2003) The Gromov norm of the Kaehler class and the Maslov index. Asian J. Math. 7(2): 269–295MathSciNetzbMATHGoogle Scholar
  8. 8.
    Domic A., Toledo D. (1987) The Gromov norm of the Kähler class of symmetric domains. Math. Ann. 276(3): 425–432CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Douady, A. L’espace de Teichmüller. Travaux de Thurston sur les surfaces, Asterisque 66-67, Société de Mathématique de France, pp. 127–137 (1979)Google Scholar
  10. 10.
    Farb, B., Margalit, D. A primer on mapping class groups. In preparationGoogle Scholar
  11. 11.
    Goldman, W.M. Mapping class group dynamics on surface group representations. In: Problems in Mapping Class Groups and Related Topics. Proceedings of Symposia in Pure Math. Amer. Math. Soc. (to appear)Google Scholar
  12. 12.
    Goldman, W.M. Discontinuous groups and the Euler Class. Thesis, University of California at Berkeley (1980)Google Scholar
  13. 13.
    Goldman W.M. (1988) Topological components of spaces of representations. Invent. Math. 93(3): 557–607CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Gothen P.B. (2001) Components of spaces of representations and stable triples. Topology 40(4): 823–850CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Hernàndez Lamoneda L. (1991) Maximal representations of surface groups in bounded symmetric domains. Trans. Amer. Math. Soc. 324, 405–420CrossRefMathSciNetGoogle Scholar
  16. 16.
    Ivanov, N.V. Mapping Class Groups. Handbook of Geometric Topology. North-Holland, Amsterdam, pp. 523–633 (2002)Google Scholar
  17. 17.
    Korevaar N.J., Schoen R.M. (1993) Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1(3–4): 561–659MathSciNetzbMATHGoogle Scholar
  18. 18.
    Labourie, F. Anosov flows, surface groups and curves in projective space. Invent. Math. (to appear), arXiv:math.DG/0401230Google Scholar
  19. 19.
    Labourie, F. Cross Ratios, Anosov Representations and the Energy Functional on Teichmüller space. Preprint arXiv:math.DG/0512070Google Scholar
  20. 20.
    Labourie, F. Crossratios, Surface Groups, \({SL}(n,\mathbb{R})\) and \({C}^{1,h}({S}^1)\rtimes{D}iff^h({S}^1)\). Preprint, arXiv:math.DG/0502441Google Scholar
  21. 21.
    Gothen, P., García-Prada, O., Mundet i Riera, I. Connected components of the representation variety for Sp \((2n,{\mathbb R})\), Preprint in preparationGoogle Scholar
  22. 22.
    Satake I. (1980) Algebraic Structures of Symmetric Domains. Kanô Memorial Lectures, vol. 4, Iwanami Shoten, TokyozbMATHGoogle Scholar
  23. 23.
    Toledo D. (1989) Representations of surface groups in complex hyperbolic space. J. Differential. Geom. 29(1): 125–133MathSciNetzbMATHGoogle Scholar
  24. 24.
    Wienhard, A. Bounded Cohomology and Geometry. Ph.D. thesis, Universität Bonn, Bonner Mathematische Schriften Nr. 368 (2004)Google Scholar
  25. 25.
    Wienhard A. (2004) A generalisation of Teichmüller space in the Hermitian context. Séminaire de Théorie Spectrale et Géométrie Grenoble 22, 103–123MathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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