The symplectic nature of fundamental groups of surfaces. (English) Zbl 0574.32032

A natural symplectic structure on the space of representations \(\pi\) \(\to G\), where \(\pi\) is the fundamental group of a Riemann surface and G is a semisimple Lie group, is introduced. The author considers this result as a common cause for the existence of a symplectic structure on different moduli spaces associated with Riemann surfaces. For example if \(G=PSL_ 2({\mathbb{R}})\) then the symplectic structure coincides with the Weil- Petersson’s one on Teichmüller space.
Reviewer: A.Givental’


32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H30 Coverings of curves, fundamental group
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
30F99 Riemann surfaces
14E20 Coverings in algebraic geometry
32G13 Complex-analytic moduli problems
22E46 Semisimple Lie groups and their representations
14H15 Families, moduli of curves (analytic)
Full Text: DOI


[1] Ahlfors, L., Some remarks on Teichmüller’s space of Riemann surfaces, Ann. of Math., 74, 171-191 (1961) · Zbl 0146.30602
[2] Ahlfors, L., Lectures on Quasiconformal Mappings (1966), Van Nostrand: Van Nostrand New York · Zbl 0138.06002
[3] Atiyah, M.; Bott, R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London A, 308, 523-615 (1982) · Zbl 0509.14014
[4] Brown, K., Cohomology of groups, (Graduate Texts in Mathematics No. 87 (1982), Springer-Verlag: Springer-Verlag New York) · Zbl 0367.18012
[5] Bers, L., Uniformization, moduli and Kleinian groups, Bull. London Math. Soc., 4, 257-300 (1972) · Zbl 0257.32012
[6] Birman, J., Braids, links and mapping class groups, Ann. of Math. Stud., 82 (1974)
[7] Earle, C., Teichmüller theory, (Harvey, W., Discrete Groups and Automorphism Functions (1977), Academic Press: Academic Press New York), 143-162
[8] Fox, R., The free differential calculus. I. Derivations in the group ring, Ann. of Math., 57, 547-560 (1953) · Zbl 0050.25602
[9] Goldman, W., Discontinuous Groups and the Euler Class, (Thesis (1980), University of California: University of California Berkeley)
[10] Goldman, W., Characteristic classes and representations of discrete subgroups of Lie groups, Bull. Amer. Math. Soc. (N. S), 6, 91-94 (1982) · Zbl 0493.57011
[11] Goldman, W., Representations of fundamental groups of surfaces, (Proceedings of the Special Year in Topology. Proceedings of the Special Year in Topology, Maryland. Proceedings of the Special Year in Topology. Proceedings of the Special Year in Topology, Maryland, Lecture Notes in Mathematics (1983-1984), Springer-Verlag: Springer-Verlag New York), to appear
[14] Gunning, R. C., Lectures on Vector Bundles over Riemann Surfaces (1967), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J · Zbl 0163.31903
[15] Hejhal, D., Monodromy groups and Poincaré series, Bull. Amer. Math. Soc., 84, 339-376 (1978) · Zbl 0393.30035
[16] Hopf, H., Fundamentalgruppe und zweite Bettische gruppe, Comm. Math. Helv., 14, 257-309 (1942) · JFM 68.0503.01
[18] Kodaira, K.; Spencer, D. C., On deformations of complex analytic structures, I, 71, 43-76 (1960) · Zbl 0128.16902
[19] Kra, I., Automorphic Forms and Kleinian Groups (1972), Benjamin: Benjamin Reading, Mass · Zbl 0253.30015
[20] Lyndon, R., Cohomology theory of groups with one defining relation, Ann. of Math., 52, 650-665 (1950) · Zbl 0039.02302
[21] Marsden, J.; Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5, 121-130 (1974) · Zbl 0327.58005
[22] Milnor, J., Construction of universal bundles, I, Ann. of Math., 63, 272-284 (1958) · Zbl 0071.17302
[24] Narasimhan, M. S.; Seshadri, C. S., Holomorphic vector bundles on a compact Riemann surface, Math. Ann., 155, 69-80 (1964) · Zbl 0122.16701
[25] Narasimhan, M. S.; Seshadri, C. S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math., 82, 540-567 (1965) · Zbl 0171.04803
[26] Nijenhuis, A.; Richardson, R., Deformations of homomorphisms of Lie groups and Lie algebras, Bull. Amer. Math. Soc., 73, 175 (1967) · Zbl 0153.04402
[27] Raghunathan, M. S., Discrete Subgroups of Lie Groups (1972), Springer-Verlag: Springer-Verlag New York · Zbl 0254.22005
[28] Shimura, G., Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan, 11, 291-311 (1959) · Zbl 0090.05503
[29] Steenrod, N., The Topology of Fiber Bundles (1951), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J
[30] Weinstein, A., Lectures on Symplectic Manifolds, (C.B.M.S. 29 (1977), Amer. Math. Soc: Amer. Math. Soc Providence, R.I) · Zbl 0406.53031
[31] Wolpert, S., An elementary formula for the Fenchel-Nielsen twist, Comm. Math. Helv., 56, 132-135 (1981) · Zbl 0467.30036
[32] Wolpert, S., The Fenchel-Nielsen deformation, Ann. of Math., 115, 501-528 (1982) · Zbl 0496.30039
[33] Wolpert, S., On the symplectic geometry of deformations of a hyperbolic surface, Ann of Math., 117, 207-234 (1983) · Zbl 0518.30040
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.