Projective structures with discrete holonomy representations

Shiga, Hiroshige; Tanigawa, Harumi

doi:10.1090/S0002-9947-99-02043-7

Projective structures with discrete holonomy representations
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by Hiroshige Shiga and Harumi Tanigawa PDF

Trans. Amer. Math. Soc. 351 (1999), 813-823 Request permission

Abstract:

Let $K(X)$ denote the set of projective structures on a compact Riemann surface $X$ whose holonomy representations are discrete. We will show that each component of the interior of $K(X)$ is holomorphically equivalent to a complex submanifold of the product of Teichmüller spaces and the holonomy representation of every projective structure in the interior of $K(X)$ is a quasifuchsian group.

References

Lipman Bers, Holomorphic families of isomorphisms of Möbius groups, J. Math. Kyoto Univ. 26 (1986), no. 1, 73–76. MR 827159, DOI 10.1215/kjm/1250520965
C. J. Earle, I. Kra, and S. L. Krushkal′, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), no. 2, 927–948. MR 1214783, DOI 10.1090/S0002-9947-1994-1214783-6
D. Drasin, C. J. Earle, F. W. Gehring, I. Kra, and A. Marden (eds.), Holomorphic functions and moduli. II, Mathematical Sciences Research Institute Publications, vol. 11, Springer-Verlag, New York, 1988. MR 955827
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, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), no. 3, 297–326. MR 882826
Dennis A. Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), no. 1, 1–55. MR 463429, DOI 10.1007/BF02392015
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}
Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
Irwin Kra, A generalization of a theorem of Poincaré, Proc. Amer. Math. Soc. 27 (1971), 299–302. MR 301189, DOI 10.1090/S0002-9939-1971-0301189-7
Irwin Kra, Deformations of Fuchsian groups, Duke Math. J. 36 (1969), 537–546. MR 265582
Irwin Kra, Deformations of Fuchsian groups. II, Duke Math. J. 38 (1971), 499–508. MR 285734
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
Bernard Maskit, On a class of Kleinian groups, Ann. Acad. Sci. Fenn. Ser. A I No. 442 (1969), 8. MR 0252638
Hiroshige Shiga, Projective structures on Riemann surfaces and Kleinian groups, J. Math. Kyoto Univ. 27 (1987), no. 3, 433–438. MR 910228, DOI 10.1215/kjm/1250520657
Zbigniew Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), no. 2, 347–355. MR 1037218, DOI 10.1090/S0002-9939-1991-1037218-8
G. Reeb, Équations différentielles et analyse non classique (d’après J.-L. Callot (Oran)), Proceedings of the IV International Colloquium on Differential Geometry (Univ. Santiago de Compostela, Santiago de Compostela, 1978) Cursos Congr. Univ. Santiago de Compostela, vol. 15, Univ. Santiago de Compostela, Santiago de Compostela, 1979, pp. 240–245 (French). MR 569325
H. Tanigawa, Grafting, harmonic maps and projective structures on surfaces, J. Differential Geometry 47 (1997), 399–419.
—, Divergence of projective structures and lengths of measured laminations, (preprint).