Kleinian circle packings
type: Item
instance of: scholarly article
title: Kleinian circle packings
zbMATH Open document ID: 0833.30025
DOI: 10.1016/0040-9383(94)00049-Q
author: John R. Parker
published in: Topology
publication date: 1995-09-10T00:00:00Z
review text: Let \(\Gamma\) be a finitely generated Kleinian group so that the limit set of \(\Gamma\) is the complement of a circle packing. There are some attractive graphics in this paper illustrating this sort of limit set. One can define, in a fairly straightforward fashion, a packing exponent of this specific circle packing. This is a purely geometric quantity which does not depend on the action of the Kleinian group. The purpose of this paper is to show that this exponent is equal to the Hausdorff dimension of the limit set. It was shown by Larman that for an arbitrary circle packing the Hausdorff dimension of the residual set is at least as large as the exponent. The main part of the proof is the converse inequality. This is proved by a delicate sharpening of Ahlfors' argument, which yields an upper bound for the Hausdorff dimension of the limit set in the case where \(\Gamma\) is geometrically finite.
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 52C15
zbMATH DE Number: 794435
zbMATH Keywords: circle packing
zbMATH Keywords: Kleinian group
zbMATH Keywords: Hausdorff dimension
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 794435