Moduli spaces of local systems and higher Teichmüller theory

Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques

Volumn 103, Issue 1, 2006, Pages 1-211

  Fock, Vladimir ^a   Goncharov, Alexander ^b  

^a INSTITUTE FOR THEORETICAL AND EXPERIMENTAL PHYSICS   (Russian Federation)

^b BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33746189727     PISSN: 00738301     EISSN: 16181913     Source Type: Journal    
DOI: 10.1007/s10240-006-0039-4     Document Type: Review

References (72)

1. [BAG] 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.
2. [BD] A. A. BEILINSON and V. G. DRINFELD, Opers, math.AG/0501398.
3. [BeK] A. BERENSTEIN and D. KAZHDAN, Geometric and unipotent crystals, Geom. Funct. Anal., Special volume, part II (2000), 188-236.
4. [BeZ] 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.
5. [BFZ96] A. BERENSTEIN, S. FOMIN and A. ZELEVINSKY, Parametrizations of canonical bases and totally positive matrices, Adv. Math., 122 (1996), no. 1, 49-149.
6. [BFZ03] 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.
7. [Bers] L. BERS, Universal Teichmüller space, Analytic Methods in Mathematical Physics (Sympos., Indiana Univ., Bloomington, Ind., 1968), pp. 65-83, Gordon and Breach (1970).
8. [Bers2] L. BERS, On the boundaries of Teichmüller spaces and on Kleinian groups, Ann. Math., 91 (1970), 670-600.
9. [Bon] F. BONAHON, The geometry of Teichmüller space via geodesic currents, Invent. Math., 92 (1988), no. 1, 139-162.
10. [Bo] 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. [Bor] M. BOROVOI, Abelianization of the second nonabelian Galois cohomology, Duke Math. J., 72 (1993), 217-239.
12. 2, Publ. Math., Inst. Hautes Étud. Sci., 94 (2001), 5-85.
13. [CG] N. CHRISS and V. GINZBURG, Representation Theory and Complex Geometry, Birkhäuser Boston, Inc., Boston, MA (1997).
14. [FCh] L. O. CHEKHOV and V. V. FOCK, Quantum Teichmüller spaces, Tear. Mat. Fiz., 120 (1999), no. 3, 511-528, math.QA/9908165.
15. [C] K. CORLETTE, Flat G-bundles with canonical metrics, J. Differ. Geom., 28 (1988), 361-382.
16. [Del] P. DELIGNE, Équations différentielles à points singuliers réguliers, Springer Lect. Notes Math., vol. 163 (1970).
17. [DS] 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. [D] S. DONALDSON, Twisted harmonic maps and the self-duality equations, Proc. Land., Math. Soc., 55 (1987), 127-131.
19. [EKLV] H. ESNAULT, B. KAHN, M. LEVINE and E. VIEHWEG, The Arason invariant and mod 2 algebraic cycles, J. Amer. MA Soc., 11 (1998), no. 1, 73-118.
20. [F] V. V. FOCK, Dual Teichmüller spaces, dg-ga/9702018.
21. [FR] 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., Sac., 191 (1999), 67-86, math. QA/9802054.
22. [FG2] V. V. FOCK and A. B. GONCHAROV, Cluster ensembles, quantization and the dilogarithm, math.AG/0311245.
23. [FG3] 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. [FG4] V. V. FOCK and A. B. GONCHAROV, Dual Teichmüller and lamination spaces, to appear in the Handbook on Teichmüller theory, math.AG/0510312.
25. [FG5] V. V. FOCK and A. B. GONCHAROV, Cluster X-Varieties, Amalganations, and Poisson-Lie Groups, Progr. Math., Birkhs̈user, volume dedicated to V. G. Drinfeld, math.RT/0508408.
26. [FG6] V. V. FOCK and A. B. GONCHAROV, to appear.
27. [FZ99] S. FOMIN and A. ZELEVINSKY, Double Bruhat cells and total positivity, J. Amer. Math. Soc., 12 (1999), no. 2, 335-380, math.RA/9912128.
28. [FZI] S. FOMIN and A. ZELEVINSKY, Cluster algebras, I, J. Amer. Math. Sac., 15 (2002), no. 2, 497-529, math.RT/0104151.
29. [FZII] S. FOMIN and A. ZELEVINSKY, Cluster algebras, II: Finite type classification, Invent. Math., 154 (2003), no. 1, 63-121, math.RA/0208229.
30. [FZ3] S. FOMIN and A. ZELEVINSKY, The Laurent phenomenon. Adv. Appl. Math., 28 (2002), no. 2, 119-144, math.CO/0104241.
31. [GGL] 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;
32. A. M. GABRIELOV, I. M. GELFAND and M. V. LOSIK, Combinatorial computation of characteristic classes, I, II. (Russian), Funkts. Anal. Prilozh. no. 3, 5-26.
33. [GRr] F. R. GANTMACHER and M. G. KREIN, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, revised edition of the 1941 Russian original.
34. [GKr2] F. R. GANTMACHER, M. G. KREIN, Sur les Matrices Oscillatores, C.R. Acad. Sci. Paris, 201 (1935), AMS Chelsea Publ., Providence, RI (2002).
35. [GSV1] M. GEKHTMAN, M. SHAPIRO and A. VAINSHTEIN, Cluster algebras and Poisson geometry, Mosc. Math. J., 3 (2003), no. 3, 899-934, math.QA/0208033.
36. [GSV2] 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.
37. [Gui] O. GUICHARD, Sur les representations de groupes de surface, preprint.
38. [Gol1] W. GOLDMAN, The symplectic nature of fundamental groups of surfaces, Adv. Math., 54 (1984), no. 2, 200-225.
39. [Gol2] W. GOLDMAN, Convex real projective structures on compact surfaces, J. Differ. Geom., 31 (1990), 126-159.
40. [G1] A. B. GONCHAROV, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math., 114 (1995), no. 2, 197-318.
41. [G2] 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).
42. [G3] 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).
43. [G4] 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.
44. [G5] A. B. GONCHAROV, Polylogarithms, regulators and Arakelov motivic complexes, J. Amer. Math. Soc., 18 (2005), no. 1, 1-6; math.AG/0207036.
45. o,n, Compos. Math., 140 (2004), no. 1, 1-14, math.AG/0204102.
46. [Ha] J. HARER, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math., 84 (1986), no. 1, 157-176.
47. [H1] N. J. HITCHIN, Lie groups and Teichmüller space, Topology, 31 (1992), no. 3, 449-473.
48. [H2] N.J. HITCHIN, The self-duality equation on a Riemann surface, Proc. Lond. Math. Soc., 55 (1987), 59-126.
49. [K] R. M. KASHAEV, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys., 43 (1998), no. 2, 105-115.
50. [Kr] 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).
51. [Ko] M. KONTSEVICH, Formal (non)commutative symplectic geometry, The Gelfand Mathematical Seminars 1990-1992, Birkhäuser Boston, Boston, MA (1993), 173-187.
52. [Lab] F. LABOURIE, Anosov flows, surface groups and curves in projective spaces, preprint, Dec. 8 (2003).
53. [L1] G. LUSZTIG, Total positivity in reductive groups, Lie Theory and Geometry, Progr. Math., vol. 123, pp. 531-568, Birkhäuser Boston, Boston, MA (1994).
54. [L2] G. LUSZTEG, 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).
55. [MM] C. McMULLEN, Iteration on Teichmüller space, Invent. Math., 99 (1990), no. 2, 425-454.
56. [Mi] J. MILNOR, Introduction to algebraic K-theory, Annals of Mathematics Studies, no. 72. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo (1971).
57. [NZ] I. NIKOLAEV and E. ZHUZHOMA, Flows on 2-dimensional manifolds, Springer Lect. Notes Math., vol. 1705 (1999).
58. [P1] R. C. PENNER, The decorated Teichmüller space of punctured surfaces, Commun. Math., Phys., 113 (1987), no. 2, 299-339.
59. [P2] R. C. PENNER, Weil-Petersson volumes, J. Differ. Geom., 35 (1992), no. 3, 559-608.
60. [P3] R. C. PENNER, Universal constructions in Teichmüller theory, Adv. Math., 98 (1993), no. 2, 143-215.
61. [P4] 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).
62. [PH] R. C. PENNER and J. L. HARER, Combinatorics of train tracks, Ann. Math. Studies, 125, Princeton University Press, Princeton, NJ (1992).
63. [Sch] I. J. SCHOENBERG, Convex domains and linear combinations of continuous functions, Bull. Amer. Math. Soc., 39 (1933), 273-280.
64. [Sch1] I. J. SCHOENBERG, Über variationsvermindernde lineare Transformationen, Math. Z, 32 (1930), 321-322.
65. [S] C. SIMPSON, Constructing variations of Hodge structures using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc., 1 (1988), 867-918.
66. [Se] 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).
67. [Str] K. STREBEL, Quadratic Differentials, Springer, Berlin, Heidelberg, New York (1984).
68. [SZ] 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.
69. n, characteristic classes and Manor K-theory, Algebraic Geometry and its Applications, Tr. Mat. Inst. Steklova,' 165 (1984), 188-204.
70. [Th] W. THURSTON, The geometry and topology of three-manifolds, Princeton University Notes, http://www.msri.org/publications/books/gt3m.
71. [W] A. M. WHITNEY, A reduction theorem for totally positive matrices, J. Anal. Math., 2 (1952), 88-92.
72. [Wo] S. WOLPERT, Geometry of the Weil-Petersson completion of the Teichmüler space, Sum Differ. Ceom., Suppl. J. Differ. Geom., VIII (2002), 357-393.