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Fox calculus, symplectic forms,
and moduli spaces


Author: Valentino Zocca
Journal: Trans. Amer. Math. Soc. 350 (1998), 1429-1466
MSC (1991): Primary 58D27
DOI: https://doi.org/10.1090/S0002-9947-98-02052-2
Correction: Tran. Amer. Math. Soc. 350 (1998), no. 8, 3407.
MathSciNet review: 1443899
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Abstract | References | Similar Articles | Additional Information

Abstract: An ``open pre-symplectic form'' on surfaces with boundary and glueing formulae are provided to symplectically integrate the symplectic form on the deformation space of representations of the fundamental group of a Riemann surface into a reductive Lie group $G$.


References [Enhancements On Off] (What's this?)

  • [AB82] M.F. Atiyah and R. Bott. The Yang-Mills-Equations over Riemann Surfaces. Philosophical Transactions of the Real Society of London, 308:523-615, 1982. MR 85k:14006
  • [AM95] A.Y. Alekseev and A.Z. Malkin. Symplectic Structure of the Moduli Space of Flat Connection on a Riemann Surface. Comm. Math. Phy. 169 (1995), 99-119. MR 96m:58028
  • [Arm83] M.A. Armstrong. Basic Topology. Springer-Verlag, 1983. MR 84f:55001
  • [BK53] H. Busemann and P.J. Kelly. Projective Geometry and Projective Metrics. Academic Press, 1953. MR 14:1008e
  • [CG93] S. Choi and W.M. Goldman. Convex Real Projective Structures on Closed Surfaces are Closed. Proceedings of the American Mathematical Society, 118(2):657-661, 1993. MR 93g:57017
  • [Fox53] R. Fox. The Free Differential Calculus. I: Derivations in the Group Ring. Annals of Mathematics, 57:547-560, 1953. MR 14:843d
  • [Gie80] D. Gieseker. Some Applications of Geometric Invariant Theory to Moduli Problems. Proc. Int. Congr. Math., Helsinki 1978, Vol. 2, 525-528, 1980. MR 81e:14007
  • [GM91] J. Gilman and B. Maskit. An Algorithm for 2-Generator Fuchsian Groups. Michigan Mathematical Journal, 38:13-31, 1991. MR 92f:30062
  • [GM88] W.M. Goldman and J.J. Millson. The Deformation Theory of Representations of Fundamental Groups of Compact Kaehler Manifolds. Publ. Math. I.H.E.S., 67:43-96, 1988. MR 90b:32041
  • [Gol84] W.M. Goldman. The Symplectic Nature of Fundamental Groups of Riemann Surfaces. Advances in Mathematics, 54:200-225, 1984. MR 86i:32042
  • [Gol86] W.M. Goldman. Invariant Functions on Lie Groups and Hamiltonian Flows of Surface Group Representations. Inventiones Mathematicae, 85:263-302, 1986. MR 87j:32069
  • [Gol90] W.M. Goldman. Convex Real Projective Structures on a Riemann Surface. Journal of Differential Geometry, 31:791-845, 1990. MR 91b:57001
  • [Gol93] W.M. Goldman. Complex Hyperbolic Kleinian Groups. Complex Geometry (Osaka, 1990), Lecture Notes Pure Appl. Math., vol. 143, Marcel Dekker, New York, 1993, pp. 31-52. MR 93k:32053
  • [Gun66] R.C. Gunning. Lectures on Riemann Surfaces. Princeton University Press, 1966. MR 34:7789
  • [Har77] W.J. Harvey, editor. Discrete Groups and Automorphic Forms. Academic Press, 1977. MR 57:6413
  • [Hit92] N.J. Hitchin. Lie Groups and Teichmüller Space. Topology, 31(3):449-473, 1992. MR 93e:32023
  • [Hop42] H. Hopf. Fundamentalgruppe und Zweite Bettische Gruppe. Commentari Mathematicae Helveticae, 14:257-309, 1942. MR 3:316e
  • [HJ94] J. Huebschmann and L.C. Jeffrey. Group Cohomology Construction of Symplectic Forms on Certain Moduli Spaces. Internat. Math. Res. Notices 1994, no. 6, 245 ff. (electronic) MR 95e:58033
  • [JM84] D. Johnson and J. Millson. Deformation Spaces of Compact Hyperbolic Manifolds. Discrete Groups in Geometry and Analysis, Proceedings of a Conference held at Yale University in honor of G.D. Mostow on his sixtieth birthday, March 1984, Birkhäuser, Boston, MA, 1987, pp. 48-106. MR 88j:22010
  • [Kar92] Y. Karshon. An Algebraic Proof for the Symplectic Structure of Moduli Space. Proceedings American Mathematical Society, 116:591-605, 1992. MR 93a:14010
  • [KM96] M. Kapovich and J.J. Millson. The Symplectic Geometry of Polygons in Euclidean Space. J. Differential Geom. 44 (1996), 479-513. CMP 97:07
  • [Kob82] S. Kobayashi. Invariant Distances for Projective Structures. Istituto Nazionale di Alta Matematica Symposium Mathematicae, 26:153-161, 1982. MR 83i:53037
  • [Kob84] S. Kobayashi. Projectively Invariant Distances for Affine and Projective Structures. Differential geometry, Banach Center Publications, 12:127-152, 1984. MR 89k:53043
  • [Lyn50] R.C. Lyndon. Cohomology Theory of Groups with a Single Defining Relation. Annals of Mathematics, 52(3):650-665, 1950. MR 13:819b
  • [MF82] D. Mumford and J. Fogarty. Geometric Invariant Theory. Springer-Verlag, 1982. MR 86a:14006
  • [Mil88] J.J. Millson. Deformations of Representations of Finitely Generated Groups. Contemporary Mathematics, 74:237-253, 1988. MR 90c:22034
  • [NS65] M.S. Narasimhan and C.S. Seshadri. Stable and Unitary Vector Bundles on a Compact Riemann Surface. Annals of Mathematics, 82:540-567, 1965. MR 32:1725
  • [Red94] D. Redfern. The Maple Handbook. Springer-Verlag, 1994.
  • [Thu] W. Thurston. The Geometry and Topology of 3-Manifolds. Lecture Notes.
  • [Wel80] R.O. Wells. Differential Analysis on Complex Manifolds. Springer-Verlag, 1980. MR 83f:58001
  • [Wol82] S.A. Wolpert. The Fenchel-Nielsen Deformation. Annals of Mathematics, 115:501-528, 1982. MR 83g:32024
  • [Wol83] S.A. Wolpert. On the Symplectic Geometry of Deformations of a Hyperbolic Surface. Annals of Mathematics, 117:207-234, 1983. MR 85e:32028
  • [Wol85] S.A. Wolpert. On the Weil-Petersson Geometry of the Moduli Space of Curves. American Journal of Mathematics, 107:969-997, 1985. MR 87b:32040
  • [Wol88] S.A. Wolpert. Cut and Paste Deformations of Riemann Surfaces. Annales Academiae Scientiarum Fennicae, Ser. A I Math. 13:401-413, 1988. MR 90e:32026
  • [Wol89] S.A. Wolpert. Riemann Surfaces, Moduli and Hyperbolic Geometry. In M. Cornalba, X. Gomez-Mont, and A. Verjovsky, editors, Lectures on Riemann Surfaces, Trieste, Italy, 1987. International Centre of Theoretical Physics, World Scientific, 1989, pp. 48-98. MR 91i:32020
  • [Zoc95] V. Zocca. Integration of the Symplectic Form on the Space of Convex Real Projective Structures on a Riemann Surface. Doctoral Dissertation, University of Maryland, 1995.

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Additional Information

Valentino Zocca
Affiliation: Mathématiques, bât. 425, Université Paris XI, F-91405 Orsay Cedex, France
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: anz@math.umd.edu

DOI: https://doi.org/10.1090/S0002-9947-98-02052-2
Received by editor(s): May 15, 1996
Additional Notes: Supported by a C.N.R. grant.
Article copyright: © Copyright 1998 American Mathematical Society

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