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Horocyclic coordinates for Riemann surfaces and moduli spaces. I. Teichmüller and Riemann spaces of Kleinian groups

Author(s): Irwin Kra
Journal: J. Amer. Math. Soc. 3 (1990), 499-578.
MSC: Primary 32G15; Secondary 30F35, 30F60
MathSciNet review: 1049503
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References:

[A1]: L. V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces, in Analytic Functions (R. Nevanlinna et. al., eds.), Princeton Univ. Press, Princeton, NJ, 1960, pp. 45-66. MR 0124486 (23:A1798)
[A2]: -, The modular function and geometric properties of quasiconformal mappings, Proc. Minn. Conf. on Complex Analysis, Springer, 1965, pp. 296-300. MR 0176064 (31:339)
[AB]: L. V. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385-404. MR 0115006 (22:5813)
[B1]: L. Bers, Spaces of Riemann surfaces, Proc. Internat. Congr. Math. (Edinburgh, 1958), Cambridge, 1960, pp. 349-361. MR 0124484 (23:A1796)
[B2]: -, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94-97. MR 0111834 (22:2694)
[B3]: -, Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215-228. MR 0132175 (24:A2022)
[B4]: -, A non-standard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1966), 113-134. MR 0192046 (33:273)
[B5]: -, Spaces of Kleinian groups, in Several Complex Variables, Maryland 1970, Lecture Notes in Math., vol. 155, Springer, Berlin, 1970, pp. 9-34. MR 0271333 (42:6216)
[B6]: -, Fiber spaces over Teichmüller spaces, Acta Math. 130 (1973), 89-126. MR 0430318 (55:3323)
[B7]: -, On spaces of Riemann surfaces with nodes, Bull. Amer. Math. Soc. 80 (1974), 1219-1222. MR 0361165 (50:13611)
[B8]: -, Spaces of degenerating Riemann surfaces, in Discontinuous Groups and Riemann Surfaces, Ann. of Math. Stud., no. 79, Princeton Univ. Press, Princeton, NJ, 1974, pp. 43-59. MR 0361051 (50:13497)
[B9]: -, Deformations and moduli of Riemann surfaces with nodes and signatures, Math. Scand. 36 (1975), 12-16. MR 0374496 (51:10696)
[B10]: -, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), 73-98. MR 0477161 (57:16704)
[B11]: -, Finite dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 131-172. MR 621883 (82k:32050)
[BG]: L. Bers and L. Greenberg, Isomorphisms between Teichmüller spaces, in Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud., no. 66, Princeton Univ. Press, Princeton, NJ, 1971, pp. 53-79. MR 0289763 (44:6951)
[Bu]: P. Buser, The collar theorem and examples, Manuscripta Math. 25 (1978), 349-357. MR 509590 (80h:53046)
[DM]: P. Deligne and D. Mumford, The irreducibility of the space of curves of a given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75-109. MR 0262240 (41:6850)
[E]: C. J. Earle, Some intrinsic coordinates on Teichmüller spaces, Proc. Amer. Math. Soc. 83 (1981), 527-531. MR 627684 (82j:32050)
[EK1]: C. J. Earle and I. Kra, On holomorphic mappings between Teichmüller spaces, in Contributions to Analysis (L. V. Ahlfors et. al., eds.), Academic Press, New York, 1974, pp. 107-124. MR 0430319 (55:3324)
[EK2]: -, Half-canonical divisors on variable Riemann surfaces, J. Math. Kyoto Univ. 26 (1986), 39-64. MR 827157 (87d:32042)
[EKM]: C. J. Earle, I. Kra, and A. Marden, Convexity of moduli spaces (to appear).
[EM]: C. J. Earle and A. Marden, Geometric complex coordinates for Teichmüller space (to appear).
[ES]: C. J. Earle and P. L. Sipe, Families of Riemann surfaces over the punctured disk (to appear). MR 1120713 (92j:30047)
[F]: J. D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Math., vol. 362, Springer-Verlag, New York, 1973. MR 0335789 (49:569)
[G]: L. Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569-573. MR 0148620 (26:6127)
[H1]: D. A. Hejhal, On Schottky and Teichmüller spaces, Adv. in Math. 15 (1975), 133-156. MR 0357862 (50:10328)
[H2]: -, Regular -groups, degenerating Riemann surfaces, and spectral theory (to appear).
[Ko]: S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Dekker, New York, 1970. MR 0277770 (43:3503)
[K1]: I. Kra, Deformations of Fuchsian groups. II, Duke Math. J. 38 (1971), 499-508. MR 0285734 (44:2951)
[K2]: -, On spaces of Kleinian groups, Comment. Math. Helv. 47 (1972), 53-69. MR 0306485 (46:5611)
[K3]: -, On Teichmüller's theorem on the quasi-invariance of the cross ratios, Israel J. Math. 30 (1978), 152-158. MR 508260 (82a:30026)
[K4]: -, Canonical mappings between Teichmüller spaces, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 143-179. MR 598682 (82b:32036)
[K5]: -, On lifting Kleinian groups to ${\text{SL}}(2,\mathbb{C})$ , in Differential Geometry and Complex Analysis (I. Chavel and H. M. Farkas, eds.), Springer-Verlag, Berlin, Tokyo, 1985, pp. 181-193. MR 780044 (86h:30078)
[K6]: -, Non-variational global coordinates for Teichmüller spaces, in Holomorphic Functions and Moduli II, Math. Sci. Res. Inst. Publ., vol. 11, Springer, New York, 1988, pp. 221-249.
[KM1]: I. Kra and B. Maskit, The deformation space of a Kleinian group, Amer. J. Math. 103 (1981), 1065-1102. MR 630778 (83e:32026)
[KM2]: -, Bases for quadratic differentials, Comment. Math. Helv. 57 (1982), 603-626. MR 694607 (85b:30059)
[Mn1]: A. Marden, Geometrically finite Kleinian groups and their deformation spaces, Chapter 8 of Discrete Groups and Automorphic Functions (W. J. Harvey, ed.), Academic Press, London, 1977, pp. 259-293. MR 0494117 (58:13047)
[Mn2]: -, Geometric relations between homeomorphic Riemann surfaces, Bull. Amer. Math. Soc. (N.S.) 3 (1980), 1001-1017. MR 585180 (81m:30047)
[Mn3]: -, Geometric complex coordinates for Teichmüller space, in Mathematical Aspects of String Theory (S. T. Yau, ed.), World Scientific, Singapore, 1987, pp. 341-364. MR 915828
[Mt1]: B. Maskit, Self-maps of Kleinian groups, Amer. J. Math. 93 (1971), 840-856. MR 0291453 (45:544)
[Mt2]: -, Decomposition of certain Kleinian groups, Acta Math. 130 (1977), 63-82. MR 0404614 (53:8414)
[Mt3]: -, Moduli of marked Riemann surfaces, Bull. Amer. Math. Soc. 80 (1974), 773-777. MR 0346149 (49:10875)
[Mt4]: -, Isomorphisms of function groups, J. Analyse Math. 32 (1977), 63-82. MR 0463430 (57:3381)
[Mr]: H. Masur, The extension of the Weil-Petersson metric to the boundary of Teichmüller spaces, Duke Math. J. 43 (1976), 623-635. MR 0417456 (54:5506)
[Ms]: J. P. Matelski, A compactness theorem for Fuchsian groups of the second kind, Duke Math. J. 43 (1976), 829-840. MR 0432921 (55:5900)
[Ra]: H. E. Rauch, Variational methods in the problem of moduli of Riemann surfaces, in Contributions to Function Theory, Tata Institute of Fundamental Research, Bombay, 1960, pp. 17-40. MR 0132173 (24:A2020)
[Ro]: H. L. Royden, Automorphisms and isometries of Teichmüller space, in Advances in the Theory of Riemann Surfaces, Ann. of Math. Stud., no. 66, Princeton Univ. Press, Princeton, NJ, 1971, pp. 369-383. MR 0288254 (44:5452)
[W1]: S. A. Wolpert, On the homology of the moduli space of stable curves, Ann. of Math. (2) 118 (1983), 491-523. MR 727702 (86h:32036)
[W2]: -, The hyperbolic metric and the geometry of the universal curve (to appear).

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