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The Action of the Mapping Class Group on Maximal Representations

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Abstract

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(Γ g , G) in the representation variety Rep(Γ g , G). Repmax(Γ 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(Γ g , G) when \(G= {\rm Sp}(2n,\mathbb{R})\), SU(n,n), SO*(4n), Spin(2,n).

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References

  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): 111–170

    MathSciNet  MATH  Google Scholar 

  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)

  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)

  4. Burger, M., Iozzi, A., Wienhard, A. Surface group representations with maximal Toledo invariant. Preprint

  5. Burger, M., Iozzi, A., Wienhard, A. Tight embeddings. Preprint

  6. Burger M., Iozzi A., Wienhard A. (2003) Surface group representations with maximal Toledo invariant. C. R. Acad. Sci. Paris, Sér. I 336, 387–390

    MathSciNet  MATH  Google Scholar 

  7. Clerc J.L., Ørsted B. (2003) The Gromov norm of the Kaehler class and the Maslov index. Asian J. Math. 7(2): 269–295

    MathSciNet  MATH  Google Scholar 

  8. Domic A., Toledo D. (1987) The Gromov norm of the Kähler class of symmetric domains. Math. Ann. 276(3): 425–432

    Article  MathSciNet  MATH  Google Scholar 

  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)

  10. Farb, B., Margalit, D. A primer on mapping class groups. In preparation

  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)

  12. Goldman, W.M. Discontinuous groups and the Euler Class. Thesis, University of California at Berkeley (1980)

  13. Goldman W.M. (1988) Topological components of spaces of representations. Invent. Math. 93(3): 557–607

    Article  MathSciNet  MATH  Google Scholar 

  14. Gothen P.B. (2001) Components of spaces of representations and stable triples. Topology 40(4): 823–850

    Article  MathSciNet  MATH  Google Scholar 

  15. Hernàndez Lamoneda L. (1991) Maximal representations of surface groups in bounded symmetric domains. Trans. Amer. Math. Soc. 324, 405–420

    Article  MathSciNet  Google Scholar 

  16. Ivanov, N.V. Mapping Class Groups. Handbook of Geometric Topology. North-Holland, Amsterdam, pp. 523–633 (2002)

  17. Korevaar N.J., Schoen R.M. (1993) Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1(3–4): 561–659

    MathSciNet  MATH  Google Scholar 

  18. Labourie, F. Anosov flows, surface groups and curves in projective space. Invent. Math. (to appear), arXiv:math.DG/0401230

  19. Labourie, F. Cross Ratios, Anosov Representations and the Energy Functional on Teichmüller space. Preprint arXiv:math.DG/0512070

  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/0502441

  21. Gothen, P., García-Prada, O., Mundet i Riera, I. Connected components of the representation variety for Sp \((2n,{\mathbb R})\), Preprint in preparation

  22. Satake I. (1980) Algebraic Structures of Symmetric Domains. Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo

    MATH  Google Scholar 

  23. Toledo D. (1989) Representations of surface groups in complex hyperbolic space. J. Differential. Geom. 29(1): 125–133

    MathSciNet  MATH  Google Scholar 

  24. Wienhard, A. Bounded Cohomology and Geometry. Ph.D. thesis, Universität Bonn, Bonner Mathematische Schriften Nr. 368 (2004)

  25. Wienhard A. (2004) A generalisation of Teichmüller space in the Hermitian context. Séminaire de Théorie Spectrale et Géométrie Grenoble 22, 103–123

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ, 08540, USA

    Anna Wienhard

  2. Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL, 60637-1514, USA

    Anna Wienhard

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  1. Anna Wienhard
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Correspondence to Anna Wienhard.

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Wienhard, A. The Action of the Mapping Class Group on Maximal Representations. Geom Dedicata 120, 179–191 (2006). https://doi.org/10.1007/s10711-006-9079-7

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  • DOI: https://doi.org/10.1007/s10711-006-9079-7

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