Tohoku Mathematical Journal

Remarks on Hausdorff dimensions for transient limit sets of Kleinian groups

Kurt Falk and Bernd O. Stratmann

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Abstract

In this paper we study normal subgroups of Kleinian groups as well as discrepancy groups (d-groups), that are Kleinian groups for which the exponent of convergence is strictly less than the Hausdorff dimension of the limit set. We show that the limit set of a d-group always contains a range of fractal subsets, each containing the set of radial limit points and having Hausdorff dimension strictly less than the Hausdorff dimension of the whole limit set. We then consider normal subgroups $G$ of an arbitrary non-elementary Kleinian group $H$, and show that the exponent of convergence of $G$ is bounded from below by half of the exponent of convergene of $H$. Finally, we give a discussion of various examples of d-groups.

Article information

Source
Tohoku Math. J. (2), Volume 56, Number 4 (2004), 571-582.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113246751

Digital Object Identifier
doi:10.2748/tmj/1113246751

Mathematical Reviews number (MathSciNet)
MR2097162

Zentralblatt MATH identifier
1069.30070

Subjects
Primary: 30F40: Kleinian groups [See also 20H10]
Secondary: 37F35: Conformal densities and Hausdorff dimension

Keywords
Kleinian groups exponent of convergence fractal geometry

Citation

Falk, Kurt; Stratmann, Bernd O. Remarks on Hausdorff dimensions for transient limit sets of Kleinian groups. Tohoku Math. J. (2) 56 (2004), no. 4, 571--582. doi:10.2748/tmj/1113246751. https://projecteuclid.org/euclid.tmj/1113246751


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