Advertisement

All finitely generated Kleinian groups of small Hausdorff dimension are classical Schottky groups

  • Yong HouEmail author
Article
  • 14 Downloads

Abstract

This is the second part of our works on the Hausdorff dimension of Schottky groups (Hou in Geom Topol 14:473–519, 2010). In this paper we prove that there exists a universal positive number \(\lambda >0\), such that up to a finite index, any finitely-generated non-elementary Kleinian group with a limit set of Hausdorff dimension \(<\lambda \) is a classical Schottky group. The proof relies on our previous works in Hou (Geom Topol 14:473–519, 2010; J Differ Geom 57:173–195, 2001), which provide the foundation for the general result of this paper. Our results can also be considered as a converse to the well-known theorem of Doyle (Acta Math 160:249–284, 1988) and Phillips–Sarnak (Acta Math 155:173–241, 1985).

Notes

Acknowledgements

The author would like to express appreciation to Benson Farb, Peter Sarnak for the opportunity to discuss this work in detail. The author also thanks Peter Shalen, Marc Culler for their many thoughts on this paper. The author is thankful to the referee for the many corrections and suggestions on the paper. This work is dedicated to my father: Shu Ying Hou.

References

  1. 1.
    Agol, I.: Tameness of hyperbolic \(3\)-manifolds. (2004) Arxiv arXiv:math/0405568
  2. 2.
    Berdon, A.: The Geometry of Discrete Groups. Springer, Berlin (1983)CrossRefGoogle Scholar
  3. 3.
    Bishop, C., Jones, P.: Hausdorff dimension and Kleinian groups. Acta Math. 179(1), 1–39 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bowen, L.: Cheeger constants and \(L^2\)-Betti numbers. Duke Math. J. 164(3), 569–615 (2013) (to appear) Google Scholar
  5. 5.
    Bridgeman, M.: Hausdorff dimension and Weil–Petersson extension to quasifuchsian space. 14, 2 (2010)Google Scholar
  6. 6.
    Button, J.: All Fuchsian Schottky groups are classical Schottky groups. Geom. Topol. Monogr. 1, 117–125 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Calegari, D., Gabai, D.: Shrinkwrapping and the taming of hyperbolic \(3\)-manifolds. (2004) Arxiv arXiv:math/0407161
  8. 8.
    Canary, D., Taylor, E.: Kleinian groups with small limit sets. Duke Math. J. 73, 371–381 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chuckrow, V.: On Schottky groups with applications to Kleinian groups. Ann. Math. 88, 47–61 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Doyle, P.: On the bass note of a Schottky group. Acta Math. 160, 249–284 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Press, Princeton (2011)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gehring, M., Maclachlan, M.: Two-generator arithmetic Kleinian groups II. Bull. Lond. Math. Soc. 30(3), 258–266 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gromov, M.: Hyperbolic Groups. In Essays in Group Theory. In: Gersten (ed.) M.S.R.I. Publ. 8. Springer, Berlin. pp 75-263 (1987)Google Scholar
  14. 14.
    Hou, Y.: Critical exponent and displacement of negatively curved free groups. J. Differ. Geom. 57, 173–195 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hou, Y.: Kleinian groups of small Hausdorff dimension are classical Schottky groups. I. Geom. Topol. 14, 473–519 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kapovich, M.: Homological dimension and critical exponent of Kleinian groups. GAFA 18, 2017–2054 (2009)Google Scholar
  17. 17.
    Kapovich, M.: Hyperbolic Manifolds and Discrete Groups: Lecture on Thurston’s Hyperbolization, Birkhauser’s series “Progress in mathematics” (2000)Google Scholar
  18. 18.
    Marden, A.: Schottky Groups and Circles. Contributions to Analysis (a Collection of Papers Dedicated to Lipman Bers), pp. 273–278. Academic Press, New York (1974)Google Scholar
  19. 19.
    McMullen, C.T.: Hausdorff dimension and conformal dynamics, III: computation of dimension. Am. J. Math. 120(4), 691–721 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mumford, D., Series, C., Wright, D.: Indra’s Pearls. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  21. 21.
    Patterson, S.J.: Measures on Limit Sets of Kleinian Groups. Analytical and Geometrical Aspects of Hyperbolic Space, pp. 291–323. Cambrige University Press, Cambrige (1987)Google Scholar
  22. 22.
    Phillips, R., Sarnak, P.: The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups. Acta Math. 155, 173–241 (1985)Google Scholar
  23. 23.
    Sullivan, D.: Discrete conformal groups and measurable dynamics. Bull. Am. Math. Soc. 6, 57–73 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Southern University of Science and Technology of ChinaShenzhenChina

Personalised recommendations