Skip to main content
SpringerLink
Log in

Commutators, collars and the geometry of Möbius groups

  • Published:
Journal d’Analyse Mathématique Aims and scope

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

References

  1. L. V. Ahlfors,Complex Analysis, 3rd edition, McGraw-Hill, New York, 1979.

    MATH  Google Scholar 

  2. E. M. Andreev,On convex polyhedra in Lobacevskil spaces, Math. USSR Sbornik10 (1970), 413–440.

    Article  MATH  Google Scholar 

  3. A. F. Beardon,The Geometry of Discrete Groups, Springer-Verlag, Berlin, 1983.

    MATH  Google Scholar 

  4. A. F. Beardon,Iteration of Rational Functions, Springer-Verlag, Berlin, 1991.

    MATH  Google Scholar 

  5. R. Brooks and J. P. Matelski,Collars in Kleinian groups, Duke Math. J.49 (1982), 163–182.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. H. Conway and N. J. A. Sloane,Sphere Packings, Lattices and Groups, Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  7. D. Gallo,A 3-dimensional hyperbolic collar lemma, Lecture Notes in Math.971 (1983), 31–35.

    Article  MathSciNet  Google Scholar 

  8. F. W. Gehring and G. J. Martin,Stability and extremality in Jørgensen’s inequality, Complex Variables12 (1989), 277–282.

    MATH  MathSciNet  Google Scholar 

  9. F. W. Gehring and G. J. Martin,Iteration theory and inequalities for Kleinian groups, Bull. Amer. Math. Soc.21 (1989), 57–63.

    Article  MATH  MathSciNet  Google Scholar 

  10. F. W. Gehring and G. J. Martin,Inequalities for Möbius transformations and discrete groups, J. Reine Angew. Math.418 (1991), 31–76.

    MATH  MathSciNet  Google Scholar 

  11. F. W. Gehring and G. J. Martin,Axial distances in discrete Möbius groups, Proc. Natl. Acad. Sci. USA89 (1992), 1999–2001.

    Article  MATH  MathSciNet  Google Scholar 

  12. F. W. Gehring and G. J Martin,6-torsion and hyperbolic volume, Proc. Amer. Math. Soc.117 (1993), 727–735.

    Article  MATH  MathSciNet  Google Scholar 

  13. F. W. Gehring and G. J. Martin,Torsion and volume in hyperbolic 3-folds (to appear).

  14. F. W. Gehring and G. J. Martin,On the minimal volume hyperbolic 3-orbifold, Bull. Amer. Math. Soc. (to appear).

  15. F. W. Gehring and G. J. Martin,Elliptic axes in discrete groups (to appear).

  16. F. W. Gehring and G. J. Martin,Discreteness in Kleinian groups and iteration theory (to appear).

  17. J. Gilman,A geometric approach to Jørgensen’s inequality, Adv. in Math.85 (1991), 193–197.

    Article  MATH  MathSciNet  Google Scholar 

  18. N. Halpern,A proof of the collar lemma, Bull. London Math. Soc.13 (1981), 141–144.

    Article  MATH  MathSciNet  Google Scholar 

  19. R. D. Horowitz,Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math.25 (1972), 635–649.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. T. Jørgensen,Compact 3-manifolds of constant negative curvature fibering over the circle, Ann. Math.106 (1977), 61–72.

    Article  MathSciNet  Google Scholar 

  22. T. Jørgensen,Comments on a discreteness condition for subgroups of SL(2, ℂ), Can. J. Math.31 (1979), 87–92.

    Google Scholar 

  23. T. Jørgensen,Traces in 2-generator subgroups of SL(2, ℂ), Proc. Amer. Math. Soc.84 (1982), 339–343.

    Article  MathSciNet  Google Scholar 

  24. L. Keen,Collars on Riemann surfaces, inDiscontinuous Groups and Riemann Surfaces, Annals of Math. Studies79 (1974), 263–268.

    MathSciNet  Google Scholar 

  25. F. Lanner,On complexes with transitive groups of automorphisms, Medd. Lunds Univ. Math. Sem.11 (1950).

  26. A. Leutbecher,Über Spitzen diskontinuierlicher Gruppen von lineargebrochenen Transformationen, Math. Zeit.100 (1967), 183–200.

    Article  MATH  MathSciNet  Google Scholar 

  27. B. Maskit,Kleinian Groups, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  28. B. Maskit,Some special 2-generator Kleinian groups, Proc. Amer. Math. Soc.106 (1989), 175–186.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Meyerhoff,A lower bound for the volume of hyperbolic 3-orbifolds, Duke Math. J.57 (1988), 185–203.

    Article  MATH  MathSciNet  Google Scholar 

  30. B. Randol,Cylinders in Riemann surfaces, Comment. Math. Helv.54 (1979), 1–5.

    Article  MATH  MathSciNet  Google Scholar 

  31. G. Rosenberger,All generating pairs of all two-generator Fuchsian groups, Arch. Math.46 (1986), 198–204.

    Article  MATH  MathSciNet  Google Scholar 

  32. H. Shimizu,On discontinuous groups operating on the product of the upper half planes, Ann. Math.77 (1963), 33–71.

    Article  MathSciNet  Google Scholar 

  33. D. Tan,On two-generator discrete groups of Möbius transformations, Proc. Amer. Math. Soc.106 (1989), 763–770.

    Article  MATH  MathSciNet  Google Scholar 

  34. D. Tan,On inequalities for discrete groups of Möbius transformations (to appear).

  35. W. P. Thurston,The geometry and topology of three-manifolds, Princeton Univ. notes.

  36. C. R. Traina,Trace polynomial for two generator subgroups of SL(2, ℂ), Proc. Amer. Math. Soc.79 (1980), 369–372.

    Article  MathSciNet  Google Scholar 

  37. E. B. Vinberg,Discrete groups generated by reflections in Lobačevskii spaces, Math. USSR Sbornik1 (1967), 429–444.

    Article  Google Scholar 

  38. E. B. Vinberg,Some arithmetical discrete groups in Lobačevskii spaces, inDiscrete Subgroups of Lie Groups and Applications to Moduli, Oxford Univ. Press, 1975, pp.323–348.

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, University of Michigan, 48109, Ann Arbor, MI, USA

    F. W. Gehring

  2. Department of Mathematics, University of Auckland, Auckland, New Zealand

    G. J. Martin

Authors
  1. F. W. Gehring
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. J. Martin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Lars Ahlfors on his eighty-fifth birthday

This research was supported in part by grants from the U.S. National Science Foundation (FWG) and the Institut Mittag-Leffler (FWG and GJM). A portion of the first author’s 1992 Elisha Netanyahu Memorial Lecture is based on this work.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gehring, F.W., Martin, G.J. Commutators, collars and the geometry of Möbius groups. J. Anal. Math. 63, 175–219 (1994). https://doi.org/10.1007/BF03008423

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03008423

Keywords

Search

Navigation