Geometry & Topology

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

Yong Hou

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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

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

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

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Zentralblatt MATH identifier

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]

Schottky group Kleinian group Hausdorff dimension limit set


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.

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