Fox calculus, symplectic forms, and moduli spaces

Zocca, Valentino

doi:10.1090/S0002-9947-98-02052-2

Fox calculus, symplectic forms, and moduli spaces
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Trans. Amer. Math. Soc. 350 (1998), 1429-1466 Request permission

Correction: Trans. Amer. Math. Soc. 350 (1998), 3407.

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

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
A. Yu. Alekseev and A. Z. Malkin, Symplectic structure of the moduli space of flat connection[s] on a Riemann surface, Comm. Math. Phys. 169 (1995), no. 1, 99–119. MR 1328263, DOI 10.1007/BF02101598
Mark Anthony Armstrong, Basic topology, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1979 original. MR 705632, DOI 10.1007/978-1-4757-1793-8
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Suhyoung Choi and William M. Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993), no. 2, 657–661. MR 1145415, DOI 10.1090/S0002-9939-1993-1145415-8
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
D. Gieseker, Some applications of geometric invariant theory to moduli problems, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 525–528. MR 562650
J. Gilman and B. Maskit, An algorithm for $2$-generator Fuchsian groups, Michigan Math. J. 38 (1991), no. 1, 13–32. MR 1091506, DOI 10.1307/mmj/1029004258
William M. Goldman and John J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 43–96. MR 972343, DOI 10.1007/BF02699127
William M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200–225. MR 762512, DOI 10.1016/0001-8708(84)90040-9
William M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986), no. 2, 263–302. MR 846929, DOI 10.1007/BF01389091
William M. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), no. 3, 791–845. MR 1053346
William M. Goldman, Complex hyperbolic Kleinian groups, Complex geometry (Osaka, 1990) Lecture Notes in Pure and Appl. Math., vol. 143, Dekker, New York, 1993, pp. 31–52. MR 1201600
R. C. Gunning, Lectures on Riemann surfaces, Princeton Mathematical Notes, Princeton University Press, Princeton, N.J., 1966. MR 0207977
W. J. Harvey (ed.), Discrete groups and automorphic functions, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1977. MR 0466535
N. J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), no. 3, 449–473. MR 1174252, DOI 10.1016/0040-9383(92)90044-I
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Johannes Huebschmann and Lisa C. Jeffrey, Group cohomology construction of symplectic forms on certain moduli spaces, Internat. Math. Res. Notices 6 (1994), 245 ff., approx. 5 pp.}, issn=1073-7928, review= MR 1277051, doi=10.1155/S1073792894000279, DOI 10.1155/S1073792894000279
Dennis Johnson and John J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven, Conn., 1984) Progr. Math., vol. 67, Birkhäuser Boston, Boston, MA, 1987, pp. 48–106. MR 900823, DOI 10.1007/978-1-4899-6664-3_{3}
Yael Karshon, An algebraic proof for the symplectic structure of moduli space, Proc. Amer. Math. Soc. 116 (1992), no. 3, 591–605. MR 1112494, DOI 10.1090/S0002-9939-1992-1112494-2
M. Kapovich and J.J. Millson. The Symplectic Geometry of Polygons in Euclidean Space. J. Differential Geom. 44 (1996), 479–513.
Shoshichi Kobayashi, Invariant distances for projective structures, Symposia Mathematica, Vol. XXVI (Rome, 1980) Academic Press, London-New York, 1982, pp. 153–161. MR 663030
Shoshichi Kobayashi, Projectively invariant distances for affine and projective structures, Differential geometry (Warsaw, 1979) Banach Center Publ., vol. 12, PWN, Warsaw, 1984, pp. 127–152. MR 961077
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. MR 719371, DOI 10.1007/978-3-642-96676-7
John J. Millson, Deformations of representations of finitely generated groups, Geometry of group representations (Boulder, CO, 1987) Contemp. Math., vol. 74, Amer. Math. Soc., Providence, RI, 1988, pp. 237–253. MR 957522, DOI 10.1090/conm/074/957522
M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 184252, DOI 10.2307/1970710
D. Redfern. The Maple Handbook. Springer-Verlag, 1994.
W. Thurston. The Geometry and Topology of 3-Manifolds. Lecture Notes.
R. O. Wells Jr., Differential analysis on complex manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 65, Springer-Verlag, New York-Berlin, 1980. MR 608414, DOI 10.1007/978-1-4757-3946-6
Scott Wolpert, The Fenchel-Nielsen deformation, Ann. of Math. (2) 115 (1982), no. 3, 501–528. MR 657237, DOI 10.2307/2007011
Scott Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. (2) 117 (1983), no. 2, 207–234. MR 690844, DOI 10.2307/2007075
Scott Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), no. 4, 969–997. MR 796909, DOI 10.2307/2374363
Scott A. Wolpert, Cut-and-paste deformations of Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), no. 3, 401–413. MR 994473, DOI 10.5186/aasfm.1988.1330
Scott A. Wolpert, Riemann surfaces, moduli and hyperbolic geometry, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 48–98. MR 1082351
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.