Geometric exponents and Kleinian groups
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}. Concerning Minkowski's dimension, for any compact \(K\), \(\dim_H (K)\leq \underline {\dim_M}(K)\), and \(\delta_{\text{Whit}} (K)\leq \overline {\dim_M} (K)\) with equality if area\((K)=0\). For a non-elementary \(G\), \(\delta_{\text{Poin}} (G) \leq \dim_H (\Lambda)\), and if area\((\Lambda) =0\) and conditions 2. are fulfilled all exponents are equal with \(\dim_H (\Lambda)= \dim_M (\Lambda)\). The proofs are based on lemmas containing other important estimates.
scientific article; zbMATH DE number 1026494