Advertisement

Moduli spaces of local systems and higher Teichmüller theory

  • Vladimir FockEmail author
  • Alexander GoncharovEmail author
Article

Abstract

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmüller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.

Keywords

Modulus Space Boundary Component Mapping Class Group Cluster Algebra Ribbon Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. Biswas, P. Ares-Gastesi and S. Govindarajan, Parabolic Higgs bundles and Teichmüller spaces for punctured surfaces, Trans. Amer. Math. Soc., 349 (1997), no. 4, 1551–1560, alg-geom/9510011.Google Scholar
  2. 2.
    A. A. Beilinson and V. G. Drinfeld, Opers, math.AG/0501398.Google Scholar
  3. 3.
    A. Berenstein and D. Kazhdan, Geometric and unipotent crystals, Geom. Funct. Anal., Special volume, part II (2000), 188–236.Google Scholar
  4. 4.
    A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive algebras, Invent. Math., 143 (2001), no. 1, 77–128, math.RT/9912012.Google Scholar
  5. 5.
    A. Berenstein, S. Fomin and A. Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math., 122 (1996), no. 1, 49–149.Google Scholar
  6. 6.
    A. Berenstein, S. Fomin and A. Zelevinsky, Cluster algebras. III: Upper bounds and double Bruhat cells, Duke Math. J., 126 (2005), no. 1, 1–52, math.RT/0305434.Google Scholar
  7. 7.
    L. Bers, Universal Teichmüller space, Analytic Methods in Mathematical Physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), pp. 65–83, Gordon and Breach (1970).Google Scholar
  8. 8.
    L. Bers, On the boundaries of Teichmüller spaces and on Kleinian groups, Ann. Math., 91 (1970), 670–600.Google Scholar
  9. 9.
    F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), no. 1, 139–162.Google Scholar
  10. 10.
    N. Bourbaki, Lie groups and Lie algebras, Chapters 4–6, translated from the 1968 French original by A. Pressley, Elements of Mathematics (Berlin), Springer, Berlin (2002).Google Scholar
  11. 11.
    M. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J., 72 (1993), 217–239.Google Scholar
  12. 12.
    J.-J Brylinsky and P. Deligne, Central extensions of reductive groups by K2, Publ. Math., Inst. Hautes Étud. Sci., 94 (2001), 5–85.Google Scholar
  13. 13.
    N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser Boston, Inc., Boston, MA (1997).Google Scholar
  14. 14.
    L. O. Chekhov and V. V. Fock, Quantum Teichmüller spaces, Teor. Mat. Fiz., 120 (1999), no. 3, 511–528, math.QA/9908165.Google Scholar
  15. 15.
    K. Corlette, Flat G-bundles with canonical metrics, J. Differ. Geom., 28 (1988), 361–382.Google Scholar
  16. 16.
    P. Deligne, Équations différentielles à points singuliers réguliers, Springer Lect. Notes Math., vol. 163 (1970).Google Scholar
  17. 17.
    V. G. Drinfeld and V. V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, Curr. Probl. Math.,24 (1984), 81–180, in Russian.Google Scholar
  18. 18.
    S. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. Lond. Math. Soc., 55 (1987), 127–131.Google Scholar
  19. 19.
    H. Esnault, B. Kahn, M. Levine and E. Viehweg, The Arason invariant and mod 2 algebraic cycles, J. Amer. Math. Soc., 11 (1998), no. 1, 73–118.Google Scholar
  20. 20.
    V. V. Fock, Dual Teichmüller spaces, dg-ga/9702018.Google Scholar
  21. 21.
    V. V. Fock and A. A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix, Transl., Ser. 2, Amer. Math. Soc., 191 (1999), 67–86, math.QA/9802054.Google Scholar
  22. 22.
    V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm, math.AG/0311245.Google Scholar
  23. 23.
    V. V. Fock and A. B. Goncharov, Moduli spaces of convex projective structures on surfaces, to appear in Adv. Math. (2006), math.AG/0405348.Google Scholar
  24. 24.
    V. V. Fock and A. B. Goncharov, Dual Teichmüller and lamination spaces, to appear in the Handbook on Teichmüller theory, math.AG/0510312.Google Scholar
  25. 25.
    V. V. Fock and A. B. Goncharov, Cluster\(\mathcal{X}\)-Varieties, Amalganations, and Poisson-Lie Groups, Progr. Math., Birkhäuser, volume dedicated to V. G. Drinfeld, math.RT/0508408.Google Scholar
  26. 26.
    V. V. Fock and A. B. Goncharov, to appear.Google Scholar
  27. 27.
    S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc., 12 (1999), no. 2, 335–380, math.RA/9912128.Google Scholar
  28. 28.
    S. Fomin and A. Zelevinsky, Cluster algebras, I, J. Amer. Math. Soc., 15 (2002), no. 2, 497–529, math.RT/0104151.Google Scholar
  29. 29.
    S. Fomin and A. Zelevinsky, Cluster algebras, II: Finite type classification, Invent. Math., 154 (2003), no. 1, 63–121, math.RA/0208229.Google Scholar
  30. 30.
    S. Fomin and A. Zelevinsky, The Laurent phenomenon. Adv. Appl. Math., 28 (2002), no. 2, 119–144, math.CO/0104241.Google Scholar
  31. 31.
    A. M. Gabrielov, I. M. Gelfand and M. V. Losik, Combinatorial computation of characteristic classes, I, II. (Russian), Funkts. Anal. Prilozh., 9 (1975), no. 2, 12–28; no. 3, 5–26.Google Scholar
  32. 32.
    F. R. Gantmacher and M. G. Krein, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, revised edition of the 1941 Russian original.Google Scholar
  33. 33.
    F. R. Gantmacher, M. G. Krein, Sur les Matrices Oscillatores, C.R. Acad. Sci. Paris, 201 (1935), AMS Chelsea Publ., Providence, RI (2002).Google Scholar
  34. 34.
    M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J., 3 (2003), no. 3, 899–934, math.QA/0208033.Google Scholar
  35. 35.
    M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Weil–Petersson forms, Duke Math. J., 127 (2005), no. 2, 291–311, math.QA/0309138.Google Scholar
  36. 36.
    O. Guichard, Sur les répresentations de groupes de surface, preprint.Google Scholar
  37. 37.
    W. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math., 54 (1984), no. 2, 200–225.Google Scholar
  38. 38.
    W. Goldman, Convex real projective structures on compact surfaces, J. Differ. Geom., 31 (1990), 126–159.Google Scholar
  39. 39.
    A. B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math., 114 (1995), no. 2, 197–318.Google Scholar
  40. 40.
    A. B. Goncharov, Polylogarithms and motivic Galois groups, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, part 2, pp. 43–96, Amer. Math. Soc., Providence, RI (1994).Google Scholar
  41. 41.
    A. B. Goncharov, Explicit Construction of Characteristic Classes, I, M. Gelfand Seminar, Adv. Soviet Math., vol. 16, part 1, pp. 169–210, Amer. Math. Soc., Providence, RI (1993).Google Scholar
  42. 42.
    A. B. Goncharov, Deninger’s conjecture of L-functions of elliptic curves at s=3. Algebraic geometry, 4. J. Math. Sci., 81 (1996), no. 3, 2631–2656, alg-geom/9512016.Google Scholar
  43. 43.
    A. B. Goncharov, Polylogarithms, regulators and Arakelov motivic complexes, J. Amer. Math. Soc., 18 (2005), no. 1, 1–6; math.AG/0207036.Google Scholar
  44. 44.
    A. B. Goncharov and Yu. I. Manin, Multiple ζ-motives and moduli spaces ℳ0,n, Compos. Math., 140 (2004), no. 1, 1–14, math.AG/0204102.Google Scholar
  45. 45.
    J. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math., 84 (1986), no. 1, 157–176.Google Scholar
  46. 46.
    N. J. Hitchin, Lie groups and Teichmüller space, Topology, 31 (1992), no. 3, 449–473.Google Scholar
  47. 47.
    N. J. Hitchin, The self-duality equation on a Riemann surface, Proc. Lond. Math. Soc., 55 (1987), 59–126.Google Scholar
  48. 48.
    R. M. Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys., 43 (1998), no. 2, 105–115.Google Scholar
  49. 49.
    I. Kra, Deformation spaces, A Crash Course on Kleinian Groups (Lectures at a Special Session, Annual Winter Meeting, Amer. Math. Soc., San Francisco, Calif., 1974), Lect. Notes Math., vol. 400, pp. 48–70, Springer, Berlin (1974).Google Scholar
  50. 50.
    M. Kontsevich, Formal (non)commutative symplectic geometry, The Gelfand Mathematical Seminars 1990–1992, Birkhäuser Boston, Boston, MA (1993), 173–187.Google Scholar
  51. 51.
    F. Labourie, Anosov flows, surface groups and curves in projective spaces, preprint, Dec. 8 (2003).Google Scholar
  52. 52.
    G. Lusztig, Total positivity in reductive groups, Lie Theory and Geometry, Progr. Math., vol. 123, pp. 531–568, Birkhäuser Boston, Boston, MA (1994).Google Scholar
  53. 53.
    G. Lusztig, Total positivity and canonical bases, Algebraic Groups and Lie Groups, Austral. Math. Soc. Lect. Ser., vol. 9, pp. 281–295, Cambridge Univ. Press, Cambridge (1997).Google Scholar
  54. 54.
    C. McMullen, Iteration on Teichmüller space, Invent. Math., 99 (1990), no. 2, 425–454.Google Scholar
  55. 55.
    J. Milnor, Introduction to algebraic K-theory, Annals of Mathematics Studies, no. 72. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo (1971).Google Scholar
  56. 56.
    I. Nikolaev and E. Zhuzhoma, Flows on 2-dimensional manifolds, Springer Lect. Notes Math., vol. 1705 (1999).Google Scholar
  57. 57.
    R. C. Penner, The decorated Teichmüller space of punctured surfaces, Commun. Math. Phys., 113 (1987), no. 2, 299–339.Google Scholar
  58. 58.
    R. C. Penner, Weil–Petersson volumes, J. Differ. Geom., 35 (1992), no. 3, 559–608.Google Scholar
  59. 59.
    R. C. Penner, Universal constructions in Teichmüller theory, Adv. Math., 98 (1993), no. 2, 143–215.Google Scholar
  60. 60.
    R. C. Penner, The universal Ptolemy group and its completions, Geometric Galois Actions, 2, 293–312, Lond. Math. Soc. Lect. Note Ser., 243, Cambridge Univ. Press (1997).Google Scholar
  61. 61.
    R. C. Penner and J. L. Harer, Combinatorics of train tracks, Ann. Math. Studies, 125, Princeton University Press, Princeton, NJ (1992).Google Scholar
  62. 62.
    I. J. Schoenberg, Convex domains and linear combinations of continuous functions, Bull. Amer. Math. Soc., 39 (1933), 273–280.Google Scholar
  63. 63.
    I. J. Schoenberg, Über variationsvermindernde lineare Transformationen, Math. Z., 32 (1930), 321–322.Google Scholar
  64. 64.
    C. Simpson, Constructing variations of Hodge structures using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc., 1 (1988), 867–918.Google Scholar
  65. 65.
    J.-P. Serre, Cohomologie Galoisienne (French), with a contribution by J.-L. Verdier, Lect. Notes Math., no. 5, 3rd edn., v+212pp., Springer, Berlin, New York (1965).Google Scholar
  66. 66.
    K. Strebel, Quadratic Differentials, Springer, Berlin, Heidelberg, New York (1984).Google Scholar
  67. 67.
    P. Sherman and A. Zelevinsky, Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Mosc. Math. J., 4 (2004), no. 4, 947–974, math.RT/0307082.Google Scholar
  68. 68.
    A. A. Suslin, Homology of GLn, characteristic classes and Milnor K-theory, Algebraic Geometry and its Applications, Tr. Mat. Inst. Steklova, 165 (1984), 188–204.Google Scholar
  69. 69.
    W. Thurston, The geometry and topology of three-manifolds, Princeton University Notes, http://www.msri.org/publications/books/gt3m.Google Scholar
  70. 70.
    A. M. Whitney, A reduction theorem for totally positive matrices, J. Anal. Math., 2 (1952), 88–92.Google Scholar
  71. 71.
    S. Wolpert, Geometry of the Weil–Petersson completion of the Teichmüller space, Surv. Differ. Geom., Suppl. J. Differ. Geom., VIII (2002), 357–393.Google Scholar

Copyright information

© Institut des Hautes Études Scientifiques and Springer-Verlag 2006

Authors and Affiliations

  1. 1.ITEPMoscowRussia
  2. 2.Department of MathematicsBrown UniversityProvidenceUSA

Personalised recommendations