Limit sets of geometrically finite free Kleinian groups
type: Item
instance of: scholarly article
title: Limit sets of geometrically finite free Kleinian groups
zbMATH Open document ID: 0537.30035
author: Tohru Akaza
author: Katsumi Inoue
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1984-01-01T00:00:00Z
review text: It is known that the Hausdorff dimension of the limit set of a geometrically finite Kleinian group is equal to its exponent of convergence. Historically the first approach towards a proof of this statement, and also the most elementary one, was found by T. Akaza. It has the disadvantage that it appears to be necessary to make some further assumptions on the nature of the group; at any rate such assumptions have had to be made in all applications of the method to date. The purpose of this paper is to give a description of the method applied to a fairly general class of groups, namely those that are free as groups. Although the authors imply in {\S} 2.2 that parabolic elements are permitted it is known that Proposition 5 of this paper is false in this case [see \textit{D. Sullivan}, Entropy, Hausdorff measure old and new, and limit sets of geometrically finite Kleinian groups (to appear in Acta Math.); Theorem 2). The reason appears to be that Proposition 1 is not correctly formulated; were this done then it would presumably follow that the main result of the paper would hold under the additional assumption that the group contains no parabolic elements.
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 3853453
zbMATH Keywords: Poincaré dimension
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: limit set
zbMATH Keywords: exponent of convergence
zbMATH Keywords: parabolic elements
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228898
OpenAlex ID: W2158247066
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cites work: Q3660064
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DOI: 10.2748/TMJ/1178228898
description: scientific article; zbMATH DE number 3853453