On monodromy of complex projective structures. (English) Zbl 0839.57011

Kapovich, Michael

Invent. Math. 119, No. 2, 243-265 (1995).

Summary: We prove that for any nonelementary representation \(\rho : \pi_1 (S) \to \text{SL} (2, \mathbb{C})\) of the fundamental group of a closed orientable hyperbolic surface \(S\) there exists a complex projective structure on \(S\) with the monodromy \(\rho\).

MSC:

57M50

General geometric structures on low-dimensional manifolds

Keywords:

representation; fundamental group; hyperbolic surface

References:

[1]	P. Appell, E. Goursat, P. Fatou: Théorie des Functions Algébraiques. Vol. 2, Paris, Gauthier-Villars 1930.
[2]	C. Banica, O. Stanasila: Algebraic methods in the global theory of complex spaces. London, New York: Wiley, 1976 · Zbl 0334.32001
[3]	J. Birman: Braids, links, and mapping class groups, Princeton University Press, 1975
[4]	S. Choi, W. Goldman: Convex real projective structures on closed surfaces are closed. Proceedings of Amercian Math. Society118 (1993), N 4, 1227-1236 · Zbl 0810.57005 · doi:10.1090/S0002-9939-1993-1136234-7
[5]	V. Chuckrow: Schottky groups with applications to Kleinian groups. Ann. Math.,88 (1968) 47-61 · Zbl 0186.40603 · doi:10.2307/1970555
[6]	A. Douady, C. Earle: Conformally natural extension of homeomorphisms of the circle. Acta Math.157 (1986) 23-48 · Zbl 0615.30005 · doi:10.1007/BF02392590
[7]	D. Gallo: Complex projective structures with prescribed monodromy. Bull. AMS,20 (1989) 31-34 · Zbl 0674.30032 · doi:10.1090/S0273-0979-1989-15687-5
[8]	D. Gallo, W. Goldman, M. Porter: Projective structures with monodromy in PSL (2, ?) (Preprint 1987)
[9]	W. Goldman: Topological components of spaces of representations. Invent. Math.93 (1988) 557-607 · Zbl 0655.57019 · doi:10.1007/BF01410200
[10]	W. Goldman: Projective structures with Fuchsian holonomy. J. Diff. Geom.25 (1987) 297-326 · Zbl 0595.57012
[11]	L. Greenberg: Homomorphisms of triangle groups in PSL (2, ?). In: Riemann surfaces and related topics. Stony Brook, Ann. Math. Stud., vol.97, 1980
[12]	R. Gunning: Special coordinate coverings of Riemann Surface. Math. Ann.170, 67-86 (1967) · Zbl 0144.33501 · doi:10.1007/BF01362287
[13]	R. Gunning: On affine and complex projective structures. In: Riemann Surfaces and Related Topics. Stony Brook, Ann. Math. Stud., vol. 97, 1980
[14]	R. Gunning: Lectures on Vector bundles over Riemann Surfaces. Princeton University Press, 1967 · Zbl 0163.31903
[15]	D. Hejhal: Monodromy groups and linearly polymorphic functions. Acta Math.135 (1975) 1-55 · Zbl 0333.34002 · doi:10.1007/BF02392015
[16]	B. Maskit: Kleinian groups, Springer Verlag, 1987 · Zbl 0627.30039
[17]	B. Maskit: On a class of Kleinian groups. Ann. Ac. Sci. Fenn., Ser. A,442 (1969) 1-8 · Zbl 0183.08901
[18]	S. Matsumoto: Foundations of flat conformal structures. Adv. Stud. Pure Appl. Math.20 (1992), Aspects of Low Dimensional Manifolds, p. 167-251 · Zbl 0816.53020
[19]	M. Narasimhan, C. Seshadri: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math.82 (1965) 540-567 · Zbl 0171.04803 · doi:10.2307/1970710
[20]	W. Thurston: Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle (Preprint)
[21]	W. Thurston: Hyperbolic structures on 3-manifolds, I, Ann. Math.124 (1986), 203-246 · Zbl 0668.57015 · doi:10.2307/1971277