Skip to main content
SpringerLink
Log in

Families of univalent functions and Kleinian groups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

The paper treats two topics. We obtain new characterizations of quasidiscs using theλ-lemma. We study regularb-groups by combining results on holomorphic families of injections with theorems on algebraic convergence of Kleinian groups and the structure of the Eichler cohomology spaces. We also obtain results about projective structures whose developing maps are coverings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Abikoff,On boundaries of Teichmüler spaces and on Kleinian groups, III, Acta Math.134 (1975), 212–237.

    Article  MathSciNet  Google Scholar 

  2. L. V. Ahlfors,Qusiconformal reflections, Acta Math.109 (1963), 291–301.

    Article  MATH  MathSciNet  Google Scholar 

  3. L. V. Ahlfors and L. Bers,Riemann’s mapping theorem for variable metrics, Ann. of Math.72 (1960), 385–404.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Bers,An approximation theorem, J. Analyse Math.14 (1965), 1–4.

    MATH  MathSciNet  Google Scholar 

  6. L. Bers,A non-standard integral equation with applications to quasiconformal mappings, Acta Math.116 (1966), 113–134.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  8. L. Bers,On boundaries of Teichmüller spaces and on Kleinian groups I, Ann. of Math.91 (1970), 570–600.

    Article  MathSciNet  Google Scholar 

  9. L. Bers,A remark on Mumford’s compactness theorem, Isr. J. Math.12 (1972), 400–407.

    Article  MATH  MathSciNet  Google Scholar 

  10. L. Bers,Finite dimensional Teichmüller spaces and generalizations, Bull. Am. Math. Soc.5 (1981), 131–172.

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Bers and H. Royden,Holomorphic families of injections, Acta Math.157 (1986), 259–286.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Douady and C. J. Earle,Conformally natural extension of homeomorphisms of the circle, Acta Math.157 (1986), 23–48.

    Article  MATH  MathSciNet  Google Scholar 

  13. F. P. Gardiner,Teichmüller Theory and Quadratic Differentials, John Wiley, New York, 1987.

    MATH  Google Scholar 

  14. F. P. Gardiner and I. Kra,Stability of Kleinian groups, Indiana Univ. J.21 (1972), 1037–1059.

    Article  MATH  MathSciNet  Google Scholar 

  15. F. Gehring,Univalent functions and the Schwarzian derivative, Comment. Math. Helv.52 (1977), 561–572.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. C. Gunning,Special coordinate coverings of Riemann surfaces, Math. Ann.170 (1967), 67–86.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. A. Hejhal,On Schottky and Koebe-like uniformizations, Duke Math. J.55 (1987), 267–286.

    Article  MATH  MathSciNet  Google Scholar 

  18. Y. Imayoshi,An analytic proof of Severi’s theorem, Complex Variables Theory Appl.2 (1983), 151–155.

    MATH  MathSciNet  Google Scholar 

  19. Y. Imayoshi,Generalizations of deFranchis theorem, Duke Math. J.50 (1983), 393–408.

    Article  MATH  MathSciNet  Google Scholar 

  20. T. Jørgensen,On discrete groups of Möbius transformations, Am. J. Math.98 (1976), 739–749.

    Article  MATH  Google Scholar 

  21. T. Jørgensen and P. Klein,Algebraic convergence of finitely generated Kleinian groups, Quart. J. Math. Oxford Ser.33 (1982), 325–332.

    Article  MATH  MathSciNet  Google Scholar 

  22. I. Kra,Deformations of Fuchsian groups, Duke Math. J.36 (1969), 537–546.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. I. Kra,Automorphic Forms and Kleinian Groups, Benjamin, Reading, Massachusetts, 1972.

    MATH  Google Scholar 

  25. I. Kra,On cohomology of Kleinian groups. IV. The Ahlfors-Sullivan construction of holomorphic Eichler integrals, J. Analyse Math.43 (1983/84), 51–87.

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. I. Kra and B. Maskit,Remarks on projective structures, inRiemann Surfaces and Related Topics, Ann. of Math. Studies97 (1981), 343–359.

    MathSciNet  Google Scholar 

  29. W. Kraus,Über den Zussamenhang einiger Charakteristiken eines einfach zusammenhängenden Bereiches mit der Kreisabbildung, Mitt. Math. Sem. Giessen21 (1932), 1–28.

    Google Scholar 

  30. S. Lang,Introduction to Differentiable Manifolds, John Wiley, New York, 1966.

    Google Scholar 

  31. O. Lehto,Quasiconformal mappings in the plane, Lecture Note, Univ. of Maryland,14 (1975).

  32. R. Mañe, P. Sad and D. Sullivan,On dynamics of rational maps, Ann. Sci. Ėcole Norm. Sup.16 (1983), 193–217.

    MATH  Google Scholar 

  33. A. Marden,The geometry of finitely generated Kleinian groups, Ann. of Math.99 (1974), 383–462.

    Article  MathSciNet  Google Scholar 

  34. B. Maskit,On boundaries of Teichmüller spaces and on Kleinian groups: II, Ann. of Math.91 (1970), 607–639.

    Article  MathSciNet  Google Scholar 

  35. B. Maskit,Self-maps of Kleinian groups, Am. J. Math.93 (1971), 840–856.

    Article  MATH  MathSciNet  Google Scholar 

  36. B. Maskit,On Klein’s combination theorem III, inAdvances in the Theory of Riemann Surfaces, Ann. of Math. Studies66 (1971), 297–316.

    MathSciNet  Google Scholar 

  37. B. Maskit,Decomposition of certain Kleinian groups, Acta Math.130 (1973), 243–263.

    Article  MATH  MathSciNet  Google Scholar 

  38. B. Maskit,Parabolic elements in Kleinian groups, Ann. of Math.,177 (1983), 659–668.

    Article  MathSciNet  Google Scholar 

  39. M. Nakada,Quasi-conformal stability of finitely generated function groups, Tôhoku Math. J.30 (1978), 45–58.

    MATH  MathSciNet  Google Scholar 

  40. C.-H. Sah,Commutator excercises, to appear.

  41. H. Shiga,Characterizations of quasi-discs and Teichmüller spaces, Tôhoku Math. J.37 (1985), 541–552.

    MATH  MathSciNet  Google Scholar 

  42. H. Shiga,Projective structure on Riemann surfaces and Kleinian groups, to appear.

  43. D. Sullivan,On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, inRiemann Surfaces and Related Topics, Ann. of Math. Studies97 (1981), 465–496.

    MathSciNet  Google Scholar 

  44. D. Sullivan,Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups, Acta Math.155 (1985), 243–260.

    Article  MATH  MathSciNet  Google Scholar 

  45. D. Sullivan and W. P. Thurston,Extending hyperbolic motions, Acta Math.157 (1986), 243–257.

    Article  MATH  MathSciNet  Google Scholar 

  46. W. P. Thurston,Zippers and schlicht functions, to appear.

Download references

Author information

Authors and Affiliations

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

    Irwin Kra

  2. Mathematical Sciences Research Institute, Berkeley, California, USA

    Irwin Kra

Authors
  1. Irwin Kra
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research partially supported by NSF grant DMS 8401280.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kra, I. Families of univalent functions and Kleinian groups. Israel J. Math. 60, 89–127 (1987). https://doi.org/10.1007/BF02766172

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02766172

Keywords

Search

Navigation