Geodesic excursions to infinity in Riemann surfaces
type: Item
instance of: scholarly article
title: Geodesic excursions to infinity in Riemann surfaces
zbMATH Open document ID: 0823.30030
DOI: 10.1007/BF03008406
author: José G. Llorente
author: J. L. Fernández
published in: Journal d'Analyse Mathématique
publication date: 1995-02-02T00:00:00Z
review text: Let \(S\) be a Riemann surface that has the unit disk \(\mathbb{D}\) as its universal covering surface: thus \(S\) is conformally equivalent to \(\mathbb{D}/G\) for some Fuchsian group \(G\). The authors prove a very interesting relation between the Hausdorff dimension of the geodesics that go to the infinity of \(S\) and the decay of the injectivity radius of \(S\). This can be expressed as follows: Let \(L_ c(G)\) be the conical (angular) limit set of the group \(G\). Then, for every \(\varepsilon> 0\), there exist \(r_ n\to 1\) and \(| z_ n|< r_ n\), \(g_ n\in G\), \(g_ n\neq id\) such that \[ d_{\mathbb{D}}(z_ n, g_ n(z_ n))< (1- r_ n)^{1-\dim\otimes \mathbb{D}\backslash L_ c(G))- \varepsilon}. \] The authors present two proofs. One uses a theorem of Littlewood about the growth of analytic functions with range in a given domain and works for the case that \(S\) is planar. The proof for the general cases uses, among other things, an idea of Sullivan. There are several remarks and examples.
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 718517
zbMATH Keywords: Poincaré metric
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: injectivity
zbMATH Keywords: Fuchsian group
zbMATH Keywords: geodesics
reviewed by: Christian Pommerenke
MaRDI profile type: Publication
cites work: Q5593095
cites work: Geometry and spectra of compact Riemann surfaces
cites work: Schwarzian derivatives, the Poincaré metric and the kernel function
cites work: The Poincaré Metric of Plane Domains
cites work: A Note on the Boundary Behaviour of Harmonic Functions
cites work: Rough isometries and the parabolicity of Riemannian manifolds
cites work: Q5835647
cites work: Function theory, random paths and covering spaces
cites work: Q4206977
cites work: Q5597066
cites work: On Bloch Functions
cites work: Q3946403
cites work: Q5597712
cites work: Symmetrization and the Poincaré metric
description: scientific article; zbMATH DE number 718517