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Horocyclic coordinates for Riemann surfaces and moduli spaces. I. Teichmüller and Riemann spaces of Kleinian groups
Author:
Irwin Kra
Journal:
J. Amer. Math. Soc. 3 (1990), 499-578
MSC:
Primary 32G15; Secondary 30F35, 30F60
MathSciNet review:
1049503
Full-text PDF Free Access
References |
Similar Articles |
Additional Information
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Clifford
J. Earle and Irwin
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(55 #3324)
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Clifford
J. Earle and Irwin
Kra, Half-canonical divisors on variable Riemann surfaces, J.
Math. Kyoto Univ. 26 (1986), no. 1, 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]
Clifford
J. Earle and Patricia
L. Sipe, Families of Riemann surfaces over the punctured disk,
Pacific J. Math. 150 (1991), no. 1, 79–96. MR 1120713
(92j:30047)
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John
D. Fay, Theta functions on Riemann surfaces, Lecture Notes in
Mathematics, Vol. 352, Springer-Verlag, Berlin, 1973. MR 0335789
(49 #569)
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Leon
Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569–573. MR 0148620
(26 #6127), http://dx.doi.org/10.1090/S0002-9904-1963-11001-0
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Dennis
A. Hejhal, On Schottky and Teichmüller spaces, Advances
in Math. 15 (1975), 133–156. MR 0357862
(50 #10328)
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-, Regular
-groups, degenerating Riemann surfaces, and spectral theory (to appear).
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Shoshichi
Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure
and Applied Mathematics, vol. 2, Marcel Dekker Inc., New York, 1970.
MR
0277770 (43 #3503)
- [K1]
Irwin
Kra, Deformations of Fuchsian groups. II, Duke Math. J.
38 (1971), 499–508. MR 0285734
(44 #2951)
- [K2]
Irwin
Kra, On spaces of Kleinian groups, Comment. Math. Helv.
47 (1972), 53–69. MR 0306485
(46 #5611)
- [K3]
Irwin
Kra, On Teichmüller’s theorem on the quasi-invariance of
cross ratios, Israel J. Math. 30 (1978),
no. 1-2, 152–158. MR 508260
(82a:30026), http://dx.doi.org/10.1007/BF02760836
- [K4]
Irwin
Kra, Canonical mappings between
Teichmüller spaces, Bull. Amer. Math. Soc.
(N.S.) 4 (1981), no. 2, 143–179. MR 598682
(82b:32036), http://dx.doi.org/10.1090/S0273-0979-1981-14870-9
- [K5]
Irwin
Kra, On lifting Kleinian groups to
𝑆𝐿(2,𝐶), Differential geometry and complex
analysis, Springer, Berlin, 1985, pp. 181–193. MR 780044
(86h:30078)
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-, 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]
Irwin
Kra and Bernard
Maskit, The deformation space of a Kleinian group, Amer. J.
Math. 103 (1981), no. 5, 1065–1102. MR 630778
(83e:32026), http://dx.doi.org/10.2307/2374258
- [KM2]
Irwin
Kra and Bernard
Maskit, Bases for quadratic differentials, Comment. Math.
Helv. 57 (1982), no. 4, 603–626. MR 694607
(85b:30059), http://dx.doi.org/10.1007/BF02565877
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A.
Marden, Geometrically finite Kleinian groups and their deformation
spaces, Discrete groups and automorphic functions (Proc. Conf.,
Cambridge, 1975), Academic Press, London, 1977, pp. 259–293. MR 0494117
(58 #13047)
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A.
Marden, Geometric relations between
homeomorphic Riemann surfaces, Bull. Amer.
Math. Soc. (N.S.) 3 (1980), no. 3, 1001–1017. MR 585180
(81m:30047), http://dx.doi.org/10.1090/S0273-0979-1980-14836-3
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Marden, Geometric complex coordinates for Teichmüller
space, Mathematical aspects of string theory (San Diego, Calif., 1986)
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1987, pp. 341–354. MR
915828
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Bernard
Maskit, Self-maps on Kleinian groups, Amer. J. Math.
93 (1971), 840–856. MR 0291453
(45 #544)
- [Mt2]
Bernard
Maskit, Decomposition of certain Kleinian groups, Acta Math.
130 (1973), 243–263. MR 0404614
(53 #8414)
- [Mt3]
Bernard
Maskit, Moduli of marked Riemann
surfaces, Bull. Amer. Math. Soc. 80 (1974), 773–777. MR 0346149
(49 #10875), http://dx.doi.org/10.1090/S0002-9904-1974-13600-1
- [Mt4]
Bernard
Maskit, Isomorphisms of function groups, J. Analyse Math.
32 (1977), 63–82. MR 0463430
(57 #3381)
- [Mr]
Howard
Masur, Extension of the Weil-Petersson metric to the boundary of
Teichmuller space, Duke Math. J. 43 (1976),
no. 3, 623–635. MR 0417456
(54 #5506)
- [Ms]
J.
Peter Matelski, A compactness theorem for Fuchsian groups of the
second kind, Duke Math. J. 43 (1976), no. 4,
829–840. MR 0432921
(55 #5900)
- [Ra]
H.
E. Rauch, Variational methods in the problem of the moduli of
Riemann surfaces., Contributions to function theory (Internat. Colloq.
Function Theory, Bombay, 1960), Tata Institute of Fundamental Research,
Bombay, 1960, pp. 17–40. MR 0132173
(24 #A2020)
- [Ro]
H.
L. Royden, Automorphisms and isometries of Teichmüller
space, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony
Brook, N.Y., 1969), Ann. of Math. Studies, No. 66. Princeton Univ. Press,
Princeton, N.J., 1971, pp. 369–383. MR 0288254
(44 #5452)
- [W1]
Scott
Wolpert, On the homology of the moduli space of stable curves,
Ann. of Math. (2) 118 (1983), no. 3, 491–523.
MR 727702
(86h:32036), http://dx.doi.org/10.2307/2006980
- [W2]
-, The hyperbolic metric and the geometry of the universal curve (to appear).
- [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
, 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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Additional Information
DOI:
http://dx.doi.org/10.1090/S0894-0347-1990-1049503-1
PII:
S 0894-0347(1990)1049503-1
Article copyright:
© Copyright 1990 American Mathematical Society
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