Geometry & Topology

Kleinian groups of small Hausdorff dimension are classical Schottky groups. I

Yong Hou

Full-text: Open access

Abstract

It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this first part of our work on this conjecture, we prove that there exists a universal positive number λ greater than 0 such that any 2–generated nonelementary Kleinian group with limit set of Hausdorff dimension less than λ is a classical Schottky group.

Article information

Source
Geom. Topol., Volume 14, Number 1 (2010), 473-519.

Dates
Received: 17 March 2008
Revised: 6 May 2009
Accepted: 23 September 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732180

Digital Object Identifier
doi:10.2140/gt.2010.14.473

Mathematical Reviews number (MathSciNet)
MR2578309

Zentralblatt MATH identifier
1188.30053

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds 57M05: Fundamental group, presentations, free differential calculus
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 37A15: General groups of measure-preserving transformations [See mainly 22Fxx]

Keywords
Schottky group Kleinian group Hausdorff dimension limit set

Citation

Hou, Yong. Kleinian groups of small Hausdorff dimension are classical Schottky groups. I. Geom. Topol. 14 (2010), no. 1, 473--519. doi:10.2140/gt.2010.14.473. https://projecteuclid.org/euclid.gt/1513732180


Export citation

References

  • I Agol, Tameness of hyperbolic $3$–manifolds
  • A F Beardon, The geometry of discrete groups, Graduate Texts in Math. 91, Springer, New York (1983)
  • P G Doyle, On the bass note of a Schottky group, Acta Math. 160 (1988) 249–284
  • Y Hou, Critical exponent and displacement of negatively curved free groups, J. Differential Geom. 57 (2001) 173–193
  • A Marden, Schottky groups and circles, from: “Contributions to analysis (a collection of papers dedicated to Lipman Bers)”, (L V Ahlfors, I Kra, B Maskit, L Nirenberg, editors), Academic Press, New York (1974) 273–278
  • D Mumford, C Series, D Wright, Indra's pearls. The vision of Felix Klein, Cambridge Univ. Press, New York (2002)
  • R S Phillips, P Sarnak, The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups, Acta Math. 155 (1985) 173–241