Remarks on Hausdorff dimensions for transient limits sets of Kleinian groups
type: Item
instance of: scholarly article
title: Remarks on Hausdorff dimensions for transient limits sets of Kleinian groups
zbMATH Open document ID: 1069.30070
author: K. Falk
author: Bernd O. Stratmann
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2005-04-15T00:00:00Z
review text: Let \(G\) be a Kleinian group acting on \(N+1\)-dimensional hyperbolic space. The authors call \(G\) a ``discrepency group'' if the Hausdorff dimension of the limit set of \(G\) strictly exceeds its critical exponent. They show that it is possible to fractionate the limit set into a family of subsets of increasing Hausdorff dimension. This is achieved by an upper bound derived from a covering argument. The authors also prove that if \(H\) is a non-elementary Kleinian group and if \(G\) is a non-trivial normal subgroup of \(H\) then the critical exponent of \(G\) is at least the half of that of \(H\). They derive this unexpected result from an observation of \textit{K. Matsuzaki} [Comp. Methods Funct. Theory 2, 469--479 (2002; Zbl 1062.30052)]. They conclude the paper with a discussion of a number of examples of deficiency groups.
reviewed by: Samuel James Patterson
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 37F35
zbMATH DE Number: 2157245
zbMATH Keywords: Kleinian groups
zbMATH Keywords: exponent of convergence
zbMATH Keywords: fractal geometry
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113246751
OpenAlex ID: W2008880142
cites work: The exponent of convergence of Poincaré series on some Kleinian groups
cites work: The Hausdorff Dimension of Singular Sets of Properly Discontinuous Groups
cites work: Inequalities for certain Fuchsian groups
cites work: Q3310101
cites work: The linear escape limit set
cites work: Hausdorff dimension and Kleinian groups
cites work: The fundamental group and the spectrum of the Laplacian
cites work: The bottom of the spectrum of a Riemannian covering.
cites work: Q3995301
cites work: Q5778153
cites work: Geodesics on quotient manifolds and their corresponding limit points
cites work: The Hausdorff dimension of the limit sets of infinitely generated Kleinian groups
cites work: Conservative action of Kleinian groups with respect to the Patterson-Sullivan measure
cites work: Q4384481
cites work: Q4328334
cites work: Q3993094
cites work: Kleinian groups with singly cusped parabolic fixed points
cites work: The exponent of convergence of Poincaré series
cites work: Some Examples of Fuchsian Groups
cites work: Further remarks on the exponent of convergence of Poincaré series
cites work: Q3751843
cites work: On the Green's function of Fuchsian groups
cites work: Q3923301
cites work: Q3926855
cites work: Q4884391
cites work: Q4662156
cites work: The density at infinity of a discrete group of hyperbolic motions
cites work: A finiteness theorem for cusps
cites work: Q3683227
cites work: Q3128538
cites work: Finitely generated fuchsian groups.
cites work: Q3687182
cites work: Q3687381
DOI: 10.2748/TMJ/1113246751
description: scientific article; zbMATH DE number 2157245