Let G be a split semisimple algebraic group over 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 , 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.
@article{PMIHES_2006__103__1_0, author = {Fock, Vladimir and Goncharov, Alexander}, title = {Moduli spaces of local systems and higher {Teichm\"uller} theory}, journal = {Publications Math\'ematiques de l'IH\'ES}, pages = {1--211}, publisher = {Springer}, volume = {103}, year = {2006}, doi = {10.1007/s10240-006-0039-4}, mrnumber = {2233852}, zbl = {1099.14025}, language = {en}, url = {http://www.numdam.org/articles/10.1007/s10240-006-0039-4/} }
TY - JOUR AU - Fock, Vladimir AU - Goncharov, Alexander TI - Moduli spaces of local systems and higher Teichmüller theory JO - Publications Mathématiques de l'IHÉS PY - 2006 SP - 1 EP - 211 VL - 103 PB - Springer UR - http://www.numdam.org/articles/10.1007/s10240-006-0039-4/ DO - 10.1007/s10240-006-0039-4 LA - en ID - PMIHES_2006__103__1_0 ER -
%0 Journal Article %A Fock, Vladimir %A Goncharov, Alexander %T Moduli spaces of local systems and higher Teichmüller theory %J Publications Mathématiques de l'IHÉS %D 2006 %P 1-211 %V 103 %I Springer %U http://www.numdam.org/articles/10.1007/s10240-006-0039-4/ %R 10.1007/s10240-006-0039-4 %G en %F PMIHES_2006__103__1_0
Fock, Vladimir; Goncharov, Alexander. Moduli spaces of local systems and higher Teichmüller theory. Publications Mathématiques de l'IHÉS, Tome 103 (2006), pp. 1-211. doi : 10.1007/s10240-006-0039-4. http://www.numdam.org/articles/10.1007/s10240-006-0039-4/
1. Parabolic Higgs bundles and Teichmüller spaces for punctured surfaces, Trans. Amer. Math. Soc., 349 (1997), no. 4, 1551-1560, alg-geom/9510011. | MR | Zbl
, and ,2. A. A. Beilinson and V. G. Drinfeld, Opers, math.AG/0501398.
3. Geometric and unipotent crystals, Geom. Funct. Anal., Special volume, part II (2000), 188-236. | MR | Zbl
and ,4. Tensor product multiplicities, canonical bases and totally positive algebras, Invent. Math., 143 (2001), no. 1, 77-128, math.RT/9912012. | MR | Zbl
and ,5. Parametrizations of canonical bases and totally positive matrices, Adv. Math., 122 (1996), no. 1, 49-149. | MR | Zbl
, and ,6. Cluster algebras. III: Upper bounds and double Bruhat cells, Duke Math. J., 126 (2005), no. 1, 1-52, math.RT/0305434. | MR
, and ,7. Universal Teichmüller space, Analytic Methods in Mathematical Physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), pp. 65-83, Gordon and Breach (1970). | MR | Zbl
,8. On the boundaries of Teichmüller spaces and on Kleinian groups, Ann. Math., 91 (1970), 670-600. | MR | Zbl
,9. The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), no. 1, 139-162. | MR | Zbl
,10. Lie groups and Lie algebras, Chapters 4-6, translated from the 1968 French original by A. Pressley, Elements of Mathematics (Berlin), Springer, Berlin (2002). | MR | Zbl
,11. Abelianization of the second nonabelian Galois cohomology, Duke Math. J., 72 (1993), 217-239. | MR | Zbl
,12. Central extensions of reductive groups by K2, Publ. Math., Inst. Hautes Étud. Sci., 94 (2001), 5-85. | Numdam | MR | Zbl
and ,13. Representation Theory and Complex Geometry, Birkhäuser Boston, Inc., Boston, MA (1997). | MR | Zbl
and ,14. Quantum Teichmüller spaces, Teor. Mat. Fiz., 120 (1999), no. 3, 511-528, math.QA/9908165. | MR | Zbl
and ,15. Flat G-bundles with canonical metrics, J. Differ. Geom., 28 (1988), 361-382. | MR | Zbl
,16. Équations différentielles à points singuliers réguliers, Springer Lect. Notes Math., vol. 163 (1970). | MR | Zbl
,17. Lie algebras and equations of Korteweg-de Vries type, Curr. Probl. Math., 24 (1984), 81-180, in Russian. | MR | Zbl
and ,18. Twisted harmonic maps and the self-duality equations, Proc. Lond. Math. Soc., 55 (1987), 127-131. | MR | Zbl
,19. The Arason invariant and mod 2 algebraic cycles, J. Amer. Math. Soc., 11 (1998), no. 1, 73-118. | MR | Zbl
, , and ,20. V. V. Fock, Dual Teichmüller spaces, dg-ga/9702018.
21. 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. | MR | Zbl
and ,22. V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm, math.AG/0311245.
23. Moduli spaces of convex projective structures on surfaces, to appear in Adv. Math. (2006), math.AG/0405348. | MR | Zbl
and ,24. Dual Teichmüller and lamination spaces, to appear in the Handbook on Teichmüller theory, math.AG/0510312. | MR
and ,25. Cluster -Varieties, Amalganations, and Poisson-Lie Groups, Progr. Math., Birkhäuser, volume dedicated to V. G. Drinfeld, math.RT/0508408. | MR
and ,26. V. V. Fock and A. B. Goncharov, to appear.
27. Double Bruhat cells and total positivity, J. Amer. Math. Soc., 12 (1999), no. 2, 335-380, math.RA/9912128. | MR | Zbl
and ,28. Cluster algebras, I, J. Amer. Math. Soc., 15 (2002), no. 2, 497-529, math.RT/0104151. | MR | Zbl
and ,29. Cluster algebras, II: Finite type classification, Invent. Math., 154 (2003), no. 1, 63-121, math.RA/0208229. | MR | Zbl
and ,30. The Laurent phenomenon. Adv. Appl. Math., 28 (2002), no. 2, 119-144, math.CO/0104241. | MR | Zbl
and ,31. Combinatorial computation of characteristic classes, I, II. (Russian), Funkts. Anal. Prilozh., 9 (1975), no. 2, 12-28; no. 3, 5-26. | MR | Zbl
, and ,32. Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, revised edition of the 1941 Russian original. | Zbl
and ,33. Sur les Matrices Oscillatores, C.R. Acad. Sci. Paris, 201 (1935), AMS Chelsea Publ., Providence, RI (2002).
, ,34. Cluster algebras and Poisson geometry, Mosc. Math. J., 3 (2003), no. 3, 899-934, math.QA/0208033. | MR | Zbl
, and ,35. Cluster algebras and Weil-Petersson forms, Duke Math. J., 127 (2005), no. 2, 291-311, math.QA/0309138. | MR | Zbl
, and ,36. O. Guichard, Sur les répresentations de groupes de surface, preprint.
37. The symplectic nature of fundamental groups of surfaces, Adv. Math., 54 (1984), no. 2, 200-225. | MR | Zbl
,38. Convex real projective structures on compact surfaces, J. Differ. Geom., 31 (1990), 126-159. | MR | Zbl
,39. Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math., 114 (1995), no. 2, 197-318. | MR | Zbl
,40. 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). | MR | Zbl
,41. 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). | MR | Zbl
,42. 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. | Zbl
,43. Polylogarithms, regulators and Arakelov motivic complexes, J. Amer. Math. Soc., 18 (2005), no. 1, 1-6; math.AG/0207036. | MR | Zbl
,44. Multiple ζ-motives and moduli spaces ℳ0,n , Compos. Math., 140 (2004), no. 1, 1-14, math.AG/0204102. | Zbl
and ,45. The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math., 84 (1986), no. 1, 157-176. | MR | Zbl
,46. Lie groups and Teichmüller space, Topology, 31 (1992), no. 3, 449-473. | MR | Zbl
,47. The self-duality equation on a Riemann surface, Proc. Lond. Math. Soc., 55 (1987), 59-126. | MR | Zbl
,48. Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys., 43 (1998), no. 2, 105-115. | MR | Zbl
,49. 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). | MR | Zbl
,50. Formal (non)commutative symplectic geometry, The Gelfand Mathematical Seminars 1990-1992, Birkhäuser Boston, Boston, MA (1993), 173-187. | MR | Zbl
,51. Anosov flows, surface groups and curves in projective spaces, preprint, Dec. 8 (2003). | MR | Zbl
,52. Total positivity in reductive groups, Lie Theory and Geometry, Progr. Math., vol. 123, pp. 531-568, Birkhäuser Boston, Boston, MA (1994). | MR | Zbl
,53. 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). | MR | Zbl
,54. Iteration on Teichmüller space, Invent. Math., 99 (1990), no. 2, 425-454. | MR | Zbl
,55. Introduction to algebraic K-theory, Annals of Mathematics Studies, no. 72. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo (1971). | MR | Zbl
,56. Flows on 2-dimensional manifolds, Springer Lect. Notes Math., vol. 1705 (1999). | MR | Zbl
and ,57. The decorated Teichmüller space of punctured surfaces, Commun. Math. Phys., 113 (1987), no. 2, 299-339. | MR | Zbl
,58. Weil-Petersson volumes, J. Differ. Geom., 35 (1992), no. 3, 559-608. | MR | Zbl
,59. Universal constructions in Teichmüller theory, Adv. Math., 98 (1993), no. 2, 143-215. | MR | Zbl
,60. The universal Ptolemy group and its completions, Geometric Galois Actions, 2, 293-312, Lond. Math. Soc. Lect. Note Ser., 243, Cambridge Univ. Press (1997). | MR | Zbl
,61. Combinatorics of train tracks, Ann. Math. Studies, 125, Princeton University Press, Princeton, NJ (1992). | MR | Zbl
and ,62. Convex domains and linear combinations of continuous functions, Bull. Amer. Math. Soc., 39 (1933), 273-280. | MR | Zbl
,63. Über variationsvermindernde lineare Transformationen, Math. Z., 32 (1930), 321-322. | JFM | MR
,64. Constructing variations of Hodge structures using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc., 1 (1988), 867-918. | MR | Zbl
,65. Cohomologie Galoisienne (French), with a contribution by J.-L. Verdier, Lect. Notes Math., no. 5, 3rd edn., v+212pp., Springer, Berlin, New York (1965). | MR | Zbl
,66. Quadratic Differentials, Springer, Berlin, Heidelberg, New York (1984). | MR | Zbl
,67. 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. | MR | Zbl
and ,68. Homology of GLn , characteristic classes and Milnor K-theory, Algebraic Geometry and its Applications, Tr. Mat. Inst. Steklova, 165 (1984), 188-204. | MR | Zbl
,69. W. Thurston, The geometry and topology of three-manifolds, Princeton University Notes, http://www.msri.org/publications/books/gt3m.
70. A reduction theorem for totally positive matrices, J. Anal. Math., 2 (1952), 88-92. | MR | Zbl
,71. Geometry of the Weil-Petersson completion of the Teichmüller space, Surv. Differ. Geom., Suppl. J. Differ. Geom., VIII (2002), 357-393. | MR | Zbl
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