# zbMATH — the first resource for mathematics

Moduli spaces of local systems and higher Teichmüller theory. (English) Zbl 1099.14025
Authors’ abstract: Let $$G$$ be a split semisimple algebraic group over $$\mathbb{Q}$$ 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 $$G(\mathbb{R})$$, 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 $$S$$ has holes, we define 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.

##### MSC:
 14H60 Vector bundles on curves and their moduli 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 14D20 Algebraic moduli problems, moduli of vector bundles 14L24 Geometric invariant theory 14L35 Classical groups (algebro-geometric aspects) 22E46 Semisimple Lie groups and their representations 20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
Full Text:
##### References:
 [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. · Zbl 0964.32011 [2] A. A. Beilinson and V. G. Drinfeld, Opers, math.AG/0501398. [3] A. Berenstein and D. Kazhdan, Geometric and unipotent crystals, Geom. Funct. Anal., Special volume, part II (2000), 188–236. · Zbl 1044.17006 [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. · Zbl 1061.17006 [5] A. Berenstein, S. Fomin and A. Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math., 122 (1996), no. 1, 49–149. · Zbl 0966.17011 [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. · Zbl 1135.16013 [7] L. Bers, Universal Teichmüller space, Analytic Methods in Mathematical Physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), pp. 65–83, Gordon and Breach (1970). [8] L. Bers, On the boundaries of Teichmüller spaces and on Kleinian groups, Ann. Math., 91 (1970), 670–600. · Zbl 0197.06001 [9] F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), no. 1, 139–162. · Zbl 0653.32022 [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). [11] M. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J., 72 (1993), 217–239. · Zbl 0849.12011 [12] J.-J Brylinsky and P. Deligne, Central extensions of reductive groups by K2, Publ. Math., Inst. Hautes Étud. Sci., 94 (2001), 5–85. · Zbl 1093.20027 [13] N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser Boston, Inc., Boston, MA (1997). · Zbl 0879.22001 [14] L. O. Chekhov and V. V. Fock, Quantum Teichmüller spaces, Teor. Mat. Fiz., 120 (1999), no. 3, 511–528, math.QA/9908165. · Zbl 0986.32007 [15] K. Corlette, Flat G-bundles with canonical metrics, J. Differ. Geom., 28 (1988), 361–382. · Zbl 0676.58007 [16] P. Deligne, Équations différentielles à points singuliers réguliers, Springer Lect. Notes Math., vol. 163 (1970). · Zbl 0244.14004 [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. [18] S. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. Lond. Math. Soc., 55 (1987), 127–131. · Zbl 0634.53046 [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. · Zbl 1025.11009 [20] V. V. Fock, Dual Teichmüller spaces, dg-ga/9702018. [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. · Zbl 0945.53050 [22] V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm, math.AG/0311245. · Zbl 1225.53070 [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. [24] V. V. Fock and A. B. Goncharov, Dual Teichmüller and lamination spaces, to appear in the Handbook on Teichmüller theory, math.AG/0510312. [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. · Zbl 1162.22014 [26] V. V. Fock and A. B. Goncharov, to appear. [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. · Zbl 0913.22011 [28] S. Fomin and A. Zelevinsky, Cluster algebras, I, J. Amer. Math. Soc., 15 (2002), no. 2, 497–529, math.RT/0104151. [29] S. Fomin and A. Zelevinsky, Cluster algebras, II: Finite type classification, Invent. Math., 154 (2003), no. 1, 63–121, math.RA/0208229. · Zbl 1054.17024 [30] S. Fomin and A. Zelevinsky, The Laurent phenomenon. Adv. Appl. Math., 28 (2002), no. 2, 119–144, math.CO/0104241. · Zbl 1012.05012 [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. [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. · Zbl 0088.25103 [33] F. R. Gantmacher, M. G. Krein, Sur les Matrices Oscillatores, C.R. Acad. Sci. Paris, 201 (1935), AMS Chelsea Publ., Providence, RI (2002). · JFM 61.0070.03 [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. · Zbl 1057.53064 [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. · Zbl 1079.53124 [36] O. Guichard, Sur les répresentations de groupes de surface, preprint. · JFM 27.0536.02 [37] W. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math., 54 (1984), no. 2, 200–225. · Zbl 0574.32032 [38] W. Goldman, Convex real projective structures on compact surfaces, J. Differ. Geom., 31 (1990), 126–159. · Zbl 0711.53033 [39] A. B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math., 114 (1995), no. 2, 197–318. · Zbl 0863.19004 [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). · Zbl 0842.11043 [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). · Zbl 0809.57016 [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. · Zbl 0867.11048 [43] A. B. Goncharov, Polylogarithms, regulators and Arakelov motivic complexes, J. Amer. Math. Soc., 18 (2005), no. 1, 1–6; math.AG/0207036. · Zbl 1104.11036 [44] A. B. Goncharov and Yu. I. Manin, Multiple {$$\zeta$$}-motives and moduli spaces 0,n, Compos. Math., 140 (2004), no. 1, 1–14, math.AG/0204102. · Zbl 1047.11063 [45] J. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math., 84 (1986), no. 1, 157–176. · Zbl 0592.57009 [46] N. J. Hitchin, Lie groups and Teichmüller space, Topology, 31 (1992), no. 3, 449–473. · Zbl 0769.32008 [47] N. J. Hitchin, The self-duality equation on a Riemann surface, Proc. Lond. Math. Soc., 55 (1987), 59–126. · Zbl 0634.53045 [48] R. M. Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys., 43 (1998), no. 2, 105–115. · Zbl 0897.57014 [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). [50] M. Kontsevich, Formal (non)commutative symplectic geometry, The Gelfand Mathematical Seminars 1990–1992, Birkhäuser Boston, Boston, MA (1993), 173–187. · Zbl 0821.58018 [51] F. Labourie, Anosov flows, surface groups and curves in projective spaces, preprint, Dec. 8 (2003). [52] G. Lusztig, Total positivity in reductive groups, Lie Theory and Geometry, Progr. Math., vol. 123, pp. 531–568, Birkhäuser Boston, Boston, MA (1994). · Zbl 0845.20034 [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). · Zbl 0890.20034 [54] C. McMullen, Iteration on Teichmüller space, Invent. Math., 99 (1990), no. 2, 425–454. · Zbl 0695.57012 [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). · Zbl 0237.18005 [56] I. Nikolaev and E. Zhuzhoma, Flows on 2-dimensional manifolds, Springer Lect. Notes Math., vol. 1705 (1999). · Zbl 1022.37027 [57] R. C. Penner, The decorated Teichmüller space of punctured surfaces, Commun. Math. Phys., 113 (1987), no. 2, 299–339. · Zbl 0642.32012 [58] R. C. Penner, Weil–Petersson volumes, J. Differ. Geom., 35 (1992), no. 3, 559–608. · Zbl 0768.32016 [59] R. C. Penner, Universal constructions in Teichmüller theory, Adv. Math., 98 (1993), no. 2, 143–215. · Zbl 0772.30040 [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). · Zbl 0983.32019 [61] R. C. Penner and J. L. Harer, Combinatorics of train tracks, Ann. Math. Studies, 125, Princeton University Press, Princeton, NJ (1992). · Zbl 0765.57001 [62] I. J. Schoenberg, Convex domains and linear combinations of continuous functions, Bull. Amer. Math. Soc., 39 (1933), 273–280. · Zbl 0007.10801 [63] I. J. Schoenberg, Über variationsvermindernde lineare Transformationen, Math. Z., 32 (1930), 321–322. · JFM 56.0106.06 [64] C. Simpson, Constructing variations of Hodge structures using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc., 1 (1988), 867–918. · Zbl 0669.58008 [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). [66] K. Strebel, Quadratic Differentials, Springer, Berlin, Heidelberg, New York (1984). [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. · Zbl 1103.16018 [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. [69] W. Thurston, The geometry and topology of three-manifolds, Princeton University Notes, http://www.msri.org/publications/books/gt3m. [70] A. M. Whitney, A reduction theorem for totally positive matrices, J. Anal. Math., 2 (1952), 88–92. · Zbl 0049.17104 [71] S. Wolpert, Geometry of the Weil–Petersson completion of the Teichmüller space, Surv. Differ. Geom., Suppl. J. Differ. Geom., VIII (2002), 357–393. · Zbl 1049.32020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.