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Quasiconformal Mappings and Analysis pp 221–251Cite as

Kleinian Groups; Eichler Cohomology and the Complex Theory

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Abstract

In this expanded version of an expository lecture delivered at the University of Michigan in honor of Fred Gehring’s seventieth birthday, I discuss the complex analytic theory of finitely generated Kleinian groups. The common theme, for all but the last section, is Eichler cohomology as developed in [29] and [30]1. We follow, however, the notation established in [35]. Our aim is to illustrate the general picture, especially by good examples that can be generalized. Full proofs are found in the articles listed in the bibliography. In order to help the reader2an appendix with some standard definitions and elementary properties has been included.

Article Footnote

1See [6] for an alternate approach.

2I am grateful to the referee for this and other suggestions that improved and clarified the exposition.

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References

  1. W. Abikoff The Real Analytic Theory of Teichmüller Space, Lecture Notes in Math., vol. 820, Springer-Verlag, 1980.

    MATH  Google Scholar 

  2. W. Abikoff The Euler characteristic and inequalities for Kleinian groups, Proc. Amer. Math. Soc. 97 (1986), 593–601.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. V. Ahlfors The complex analytic stucture of the space of closed Riemann surfaces, Analytic Functions, Princeton University Press, 1960, pp. 45–66.

    Google Scholar 

  4. L. V. Ahlfors Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429 and 87 (1965), 759.

    Google Scholar 

  5. L. V. Ahlfors Eichler integrals and Bers’ area theorem, Mich. Math. J. 15 (1968), 257–263.

    Article  MathSciNet  MATH  Google Scholar 

  6. L.V. Ahlfors The structure of a finitely generated Kleinian group, Acta Math. 122 (1969), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. V. Ahlfors Some remarks on Kleinian groups, Lars Valerian Ahlfors: Collected Papers Volume 2 1954–1979, Birkhäuser, 1982, pp. 316–319.

    Google Scholar 

  8. D. E. Barrett and J. Diller Poincaré series and holomorphic averaging, Invent. Math. 110 (1992), 23–27.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. F. Beardon The Geometry of Discrete Groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, 1983.

    Google Scholar 

  10. A. F. Beardon and B. Maskit Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Bers Spaces of Riemann surfaces, Proc. Internat. Congress Math. (Edinborough, 1958), Cambridge University Press, 1960, pp. 349–361.

    Google Scholar 

  12. L. Bers Automorphic forms and Poincaré series for infinitely generated Fachsian groups, Amer. J. Math. 87 (1965), 196–214.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Bers Inequalities for finitely generated Kleinian groups, J. d’Analyse Math. 18 (1967), 23–41.

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Bers On Ahlfors’ finiteness theorem, Amer. J. Math. 89 (1967), 1078–1082.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Bers Finite dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 131–172.

    Article  MathSciNet  MATH  Google Scholar 

  16. C. J. Earle and A. Marden Geometric complex coordinates for Teichmüller space, to appear.

    Google Scholar 

  17. M. Eichler Eine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), 267–298.

    Article  MathSciNet  MATH  Google Scholar 

  18. H. M. Farkas and Y. Kopeliovich New theta constant identities, Israel J. Math. 82 (1993), 133–140.

    Article  MathSciNet  MATH  Google Scholar 

  19. H. M. Farkas and Y. Kopeliovich New theta constant identities II, Proc. Amer. Math. Soc. 123 (1995), 1009–1020.

    MathSciNet  MATH  Google Scholar 

  20. H. M. Farkas, Y. Kopeliovich, and I. Kra Uniformization of modular, curves, Comm. Anal. Geom. 4 (1996), 207–259.

    MathSciNet  MATH  Google Scholar 

  21. H. M. Farkas and I. Kra Automorphic forms for subgroups of the modular group. II: Composite level, to appear in J. d’Analyse Math.

    Google Scholar 

  22. H. M. Farkas and I. Kra Theta constant identities with applications to combinatorial number theory, submitted for publication.

    Google Scholar 

  23. H. M. Farkas and I. Kra Automorphic forms for subgroups of the modular group, Israel J. Math. 82 (1993), 87–131.

    Article  MathSciNet  MATH  Google Scholar 

  24. L. R. Ford, Automorphic Functions, Chelsea, 1951.

    Google Scholar 

  25. F. Gardiner and I. Kra Quasiconformal stability of Kleinian groups, Indiana Univ. Math. J. 21 (1972), 1037–1059.

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Greenberg On a theorem of Ahlfors and conjugate subgroups of Kleinian groups, Amer. J. Math. 89 (1967), 56–68.

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Keen and C. Series Pleating coordinates for the Maskit embedding of the Teichmuller space of punctured tori, Topology 32 (1993), 719–749.

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Komori On the automorphic functions for Fuchsian groups of genus two, RIMS (Kyoto Univ.) preprint 963 (January, 1994.

    Google Scholar 

  29. I. Kra On cohomology of Kleinian groups, Ann. of Math. 89 (1969), 533–556.

    Article  MathSciNet  MATH  Google Scholar 

  30. I. Kra On cohomology of Kleinian groups: II, Ann. of Math 90 (1969), 575–589.

    Article  MathSciNet  Google Scholar 

  31. I. Kra Deformations of Fuchsian groups. II, Duke Math. J. 38 (1971), 499–508.

    Article  MathSciNet  MATH  Google Scholar 

  32. I. Kra Automorphic Forms and Kleinian Groups, Benjamin, 1972.

    MATH  Google Scholar 

  33. I. Kra On the vanishing of Poincaré series of rational functions, Bull. Amer Math. Soc. 8 (1983), 63–66.

    Article  MathSciNet  MATH  Google Scholar 

  34. I. Kra On the cohomology of Kleinian groups IV. The Ahlfors-Sullivan con-struction of holomorphic Eichler integrals, J. d’Analyse Math. 43 (1983/84), 51–87.

    Google Scholar 

  35. I. Kra On the vanishing of and spanning sets for Poincaré series for cusp forms, Acta Math. 153 (1984), 47–116.

    Article  MathSciNet  MATH  Google Scholar 

  36. I. Kra Cusp forms associated to loxodromic elements of Kleinian groups, Duke Math. J. 52 (1985), 587–625.

    Article  MathSciNet  MATH  Google Scholar 

  37. I. Kra, On algebraic curves (of low genus) defined by Kleinian groups, 46 (1985), 147–156.

    Google Scholar 

  38. I. Kra Uniformization, automorphic forms and accessory parameters, RIMS (Kyoto Univ.) Kokyuroku 571 (1985), 54–84.

    MathSciNet  Google Scholar 

  39. I. Kra Non-variational global coordinates for Teichmüller space, Holomor-phic Functions and Moduli II, Springer-Verlag, 1988, pp. 221–249.

    Google Scholar 

  40. I. Kra Horocyclic coordinates for Riemann surfaces and moduli spaces. I: Teichmüller and Riemann spaces of Kleinian groups, J. Amer. Math. Soc. 3 (1990), 499–578.

    MathSciNet  MATH  Google Scholar 

  41. I. Kra Cusp forms associated to rank 2 parabolic subgroups of Kleinian groups, Proc. Amer. Math. Soc. 111 (1991), 803–814.

    MathSciNet  MATH  Google Scholar 

  42. I. Kra and B. Maskit Pinched two component groups, to appear in Analysis and Topology (a volume dedicated to S. Stoilow).

    Google Scholar 

  43. B. Maskit Moduli of marked Riemann surfaces, Bull. Amer Math. Soc. 80 (1974), 773–777.

    Article  MathSciNet  MATH  Google Scholar 

  44. B. Maskit Kleinian Groups, Grundlehren der mathematischen Wissenschaften, vol. 287, Springer-Verlag, 1988.

    Google Scholar 

  45. C. McMullen Amenability, Poincaré series and quasiconformal maps, Invent. Math. 97 (1989), 95–127.

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Nakada Cohomology of finitely generated Kleinian groups with an invariant component, J. Math. Soc. Japan 28 (1976), 699–711.

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Nakada Surjectivity of the Bers, map, Tôhoku Math. J. 288 (1976), 257–266.

    Article  MathSciNet  MATH  Google Scholar 

  48. W. Parry, H1 and symmetric tensors, Duke Math. J. 52 (1985), 577–586.

    Article  MathSciNet  MATH  Google Scholar 

  49. D. C. Sengupta On cohomology of finitely generated function groups, J. d’Analyse Math. 63 (1994), 1–17.

    Article  MATH  Google Scholar 

  50. D. Sullivan A finiteness theorem for cusps, Acta Math. 147 (1981), 289–299.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. State University of New York at Stony Brook, Stony Brook, NY, 11794, USA

    Irwin Kra

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  1. Irwin Kra
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Michigan, 4074 East Hall, Ann Arbor, MI, 48109-1109, USA

    Peter Duren

  2. Department of Mathematics, University of Michigan, 525 East University Street-East Hall, Ann Arbor, MI, 48109-1109, USA

    Juha Heinonen

  3. Department of Mathematics, Stanford University, Stanford, CA, 94305-2125, USA

    Brad Osgood

  4. Department of Mathematics, University of Texas, Austin, TX, 78712-1082, USA

    Bruce Palka

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Kra, I. (1998). Kleinian Groups; Eichler Cohomology and the Complex Theory. In: Duren, P., Heinonen, J., Osgood, B., Palka, B. (eds) Quasiconformal Mappings and Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0605-7_14

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  • DOI: https://doi.org/10.1007/978-1-4612-0605-7_14

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6836-9

  • Online ISBN: 978-1-4612-0605-7

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