Skip to main content
SpringerLink
Log in

Large and Small Covers of a Hyperbolic Manifold

  • Published:
Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

The exponent of convergence of a non-elementary discrete group of hyperbolic isometries measures the Hausdorff dimension of the conical limit set. In passing to a non-trivial regular cover the resulting limit sets are point-wise equal though the exponent of convergence of the cover uniformization may be strictly less than the exponent of convergence of the base. We show in this paper that, for closed hyperbolic surfaces, the previously established lower bound of one half on the exponent of convergence of “small” regular covers is sharp but is not attained. We also consider “large” (non-regular) covers. Here large and small are descriptive of the size of the exponent of convergence. We show that a Kleinian group that uniformizes a manifold homeomorphic to a surface fibering over a circle contains a Schottky subgroup whose exponent of convergence is arbitrarily close to two.

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. Agol, I.: Criteria for virtual fibering. J. Topol. 1, 269–284 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agol, I.: Tameness of hyperbolic 3-manifolds (preprint)

  3. Bers, L.: Automorphic forms for Schottky groups. Adv. Math. 16, 332–361 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bishop, C., Jones, P.: Hausdorff dimension and Kleinian groups. Acta Math. 179, 1–39 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonahon, F.: Bouts des variétés hyperboliques de dimension 3. Ann. Math. 124(2), 71–158 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv. 56, 581–598 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brooks, R.: The bottom of the spectrum of a Riemannian cover. J. Reine Angew. Math. 357, 101–114 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Canary, R.: On the Laplacian and geometry of hyperbolic 3-manifolds. J. Differ. Geom. 36, 349–367 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Canary, R.: A covering theorem for hyperbolic 3-manifolds and its applications. Topology 35, 751–778 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Canary, R., Taylor, E.C.: Kleinian groups with small limit sets. Duke Math. J. 73, 371–381 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Canary, R., Minsky, Y., Taylor, E.C.: Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds. J. Geom. Anal. 9, 17–40 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Calegari, D., Gabai, D.: Shrinkwrapping and the taming of hyperbolic manifolds. J. Am. Math. Soc. 19, 385–446 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, vol. 115. Academic Press, San Diego (1984)

    MATH  Google Scholar 

  14. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis, pp. 195–199. Princeton University Press, Princeton (1970)

    Google Scholar 

  15. Doyle, P.: On the bass note of a Schottky group. Acta Math. 160, 249–284 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Falk, K.: A note on Myrberg points and ergodicity. Math. Scand. 96, 107–116 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Falk, K., Stratmann, B.: Remarks on Hausdorff dimensions for transient limit sets of Kleinian groups. Tohoku Math. J. 56(2), 571–582 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fernández, J., Rodríguez, J.: The exponent of convergence of Riemann surfaces. Bass Riemann surfaces. Ann. Acad. Sci. Fenn. 15, 165–183 (1990)

    MATH  Google Scholar 

  19. Greenberg, L.: Finiteness theorems for Fuchsian and Kleinian groups. In: Discrete Groups and Automorphic Functions, pp. 199–257. Academic Press, New York (1977)

    Google Scholar 

  20. Krushkal, S.L., Apanosov, B.N., Gusevskii, N.A.: Kleinian Groups and Uniformization in Examples and Problems. Translations of Mathematical Monographs, vol. 62. Am. Math. Soc., Providence (1986)

    MATH  Google Scholar 

  21. Lundh, T.: Geodesics on quotient manifolds and their corresponding limit points. Mich. Math. J. 51, 279–304 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Malcev, A.: On faithful representations of infinite groups of matrices. Mat. Sib. 8, 405–422 (1940)

    MathSciNet  Google Scholar 

  23. Malcev, A.: On faithful representations of infinite groups of matrices. Am. Math. Soc. Transl. 45, 1–8 (1965)

    MathSciNet  Google Scholar 

  24. Marden, A.: Schottky groups and circles. In: Contributions to Analysis (a Collection of Papers Dedicated to Lipman Bers), pp. 273–278. Academic Press, New York (1974)

    Google Scholar 

  25. Maskit, B.: Kleinian Groups. Springer, New York (1998)

    Google Scholar 

  26. Matsuzaki, K.: Dynamics of Kleinian groups—the Hausdorff dimension of limit sets. Sugaku 51(2), 142–160 (1999) (Translation in Selected Papers on Classical Analysis, pp. 23–44. AMS Translation Series (2), vol. 204. Am. Math. Soc., Providence, 2001)

    MathSciNet  MATH  Google Scholar 

  27. Matsuzaki, K.: Convergence of the Hausdorff dimension of the limit sets of Kleinian groups. In: The Tradition of Ahlfors and Bers: Proceedings of the First Ahlfors-Bers Colloquium. Contemporary Math., vol. 256, pp. 243–254. Am. Math. Soc., Providence (2000)

    Google Scholar 

  28. Matsuzaki, K.: Conservative action of Kleinian groups with respect to the Patterson–Sullivan measure. Comput. Methods Funct. Theory 2, 469–479 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Matsuzaki, K.: Isoperimetric constants for conservative Fuchsian groups. Kodai Math. J. 28, 292–300 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Matsuzaki, K., Taniguchi, M.: Hyperbolic Manifolds and Kleinian Groups. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  31. Nicholls, P.: The Ergodic Theory of Discrete Groups. London Math. Soc. Lecture Note Series, vol. 143. Cambridge Univ. Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  32. Patterson, S.J.: The limit set of a Fuchsian group. Acta Math. 176, 241–273 (1976)

    Article  MathSciNet  Google Scholar 

  33. Patterson, S.: Some examples of Fuchsian groups. Proc. Lond. Math. Soc. 39(3), 276–298 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  34. Patterson, S.: Further remarks on the exponent of convergence of Poincaré series. Tohoku Math. J. 35(2), 357–373 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  35. Purzitsky, N.: A cutting and pasting of noncompact polygons with applications to Fuchsian groups. Acta Math. 143, 233–250 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rees, M.: Checking ergodicity of some geodesic flows with infinite Gibbs measure. Ergod. Theory Dyn. Syst. 1, 107–133 (1981)

    MathSciNet  MATH  Google Scholar 

  37. Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. Inst. Ht. Études Sci. Publ. Math. 50, 171–202 (1979)

    Article  MATH  Google Scholar 

  38. Sullivan, D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite groups. Acta Math. 153, 259–277 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  39. Sullivan, D.: Related aspects of positivity in Riemannian geometry. J. Differ. Geom. 25, 327–351 (1987)

    MATH  Google Scholar 

  40. Taylor, E.: Geometric finiteness and the convergence of Kleinian groups. Commun. Anal. Geom. 5, 497–533 (1997)

    MATH  Google Scholar 

  41. Tukia, P.: The Hausdorff dimension of the limit set of a geometrically finite Kleinian group. Acta Math. 152, 127–140 (1985)

    Article  MathSciNet  Google Scholar 

  42. Yamamoto, H.: An example of a non-classical Schottky group. Duke Math. J. 63, 193–197 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Wesleyan University, Middletown, CT, 06459, USA

    Petra Bonfert-Taylor, Katsuhiko Matsuzaki & Edward C. Taylor

Authors
  1. Petra Bonfert-Taylor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Katsuhiko Matsuzaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edward C. Taylor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Katsuhiko Matsuzaki.

Additional information

Bonfert-Taylor and Taylor were supported in part by NSF grant 0706754.

Matsuzaki was supported in part by the Van Vleck fund.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bonfert-Taylor, P., Matsuzaki, K. & Taylor, E.C. Large and Small Covers of a Hyperbolic Manifold. J Geom Anal 22, 455–470 (2012). https://doi.org/10.1007/s12220-010-9204-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-010-9204-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation