Geometric exponents and Kleinian groups
type: Item
instance of: scholarly article
title: Geometric exponents and Kleinian groups
zbMATH Open document ID: 0876.30044
author: Christopher J. Bishop
published in: Inventiones Mathematicae
publication date: 1997-11-20T00:00:00Z
review text: Let \(G\) be a Kleinian group, \(\Lambda\) its limit set and \(\{\Omega_j\}\) an enumeration of the components of \(\Omega= S^2 \backslash \Lambda\) supposed non-empty. If \(G\) is analytically finite (a.f.), i.e. \(\Omega/G\) is a finite union of finite type surfaces. The author proves that \(\Sigma_j \text{diam} (\Omega_j)^2 <\infty\), solving thus Maskit's conjecture (diameter is taken in the spherical metric). The exponent 2 is the best possible: there are finitely generated Kleinian groups with \(\Sigma_j \text{diam} (\Omega_j)^s =\infty\) for every \(s<2\). However, the author looks for the best exponent for particular groups and defines geometric critical exponents generalizing to compact sets \(K\) in \(\mathbb{R}^2\) the Besicovitch-Taylor index on \(\mathbb{R}\). Let \(\{\Omega_j\}\) be an enumeration of the bounded components of \(\Omega= \mathbb{R}^2 \backslash K\); then if \(\text{inrad} (\Omega_j)\) is the radius of the largest disk in \(\Omega_j\), \(\delta_{\text{inrad}} (K)= \inf\{s: \Sigma_j \text{inrad} (\Omega_j)^s <\infty\}\); \(\delta_{\text{diam}} (K)= \inf\{s: \Sigma_j \text{diam} (\Omega_j)^s <\infty\}\); \(\delta_{\text{Whit}} (K)= \inf\{s: \Sigma_j \text{diam} (Q_j)^s <\infty\}\), where \(\{Q_j\}\) is a Whitney decomposition of \(\Omega\). Further \(\delta_{\text{Poin}}(G) =\inf\{s: \Sigma_{g\in G} \text{dist} (g(z_0), \Lambda)^s <\infty\}\), where \(z_0\in \Omega\) and the distance is in the spherical metric, is the Poincaré exponent. A series of deep results on these exponents are proved, e.g.: \textbf{1}. If \(G\) is a.f. and \(\Omega\) has infinitely many components, then \(\delta_{\text{Poin}} (G)= \delta_{\text{inrad}} (\Lambda)= \delta_{\text{diam}} (\Lambda)\); if, in addition, area \((\Lambda) =0\), then they are equal to \(\dim_H (\Lambda)\) -- the Hausdorff dimension of \(\Lambda\). \textbf{2}. If \(G\) is non-elementary a.f. with \(\Lambda\neq S^2\), then \(\delta_{\text{Poin}} (G)= \delta_{\text{Whit}} (\Lambda)\). \textbf{3}.…
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 1026494
reviewed by: Cabiria Andreian Cazacu
MaRDI profile type: Publication
full work available at URL: s002220050113
OpenAlex ID: W2027542378
DOI: 10.1007/S002220050113
Recommended article: Q4379881
Recommended article: Geometry of Kleinian groups and its applications
Recommended article: Q2939768
Recommended article: Geometry and dynamics of Kleinian groups
Recommended article: Geometric Finiteness of Certain Kleinian Groups
Recommended article: Q5459675
Recommended article: Q4039828
Recommended article: Algebraic and geometric convergence of Kleinian groups.
Recommended article: Recent results in the geometry of Kleinian groups
Recommended article: Q3596139
description: scientific article; zbMATH DE number 1026494