Gromov hyperbolicity of Denjoy domains through fundamental domains
type: Item
instance of: scholarly article
title: Gromov hyperbolicity of Denjoy domains through fundamental domains
zbMATH Open document ID: 1289.30219
DOI: 10.5486/PMD.2012.4838
author: Ana Portilla
author: José M. Rodríguez
author: Eva Tourís
author: Peter Hästö
published in: Publicationes Mathematicae Debrecen
publication date: 2012-09-17T00:00:00Z
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 30C99
zbMATH DE Number: 6083232
zbMATH Keywords: Denjoy domain
zbMATH Keywords: flute surface
zbMATH Keywords: Gromov hyperbolicity
zbMATH Keywords: Riemann surface
zbMATH Keywords: train
zbMATH Keywords: fundamental domain
MaRDI profile type: Publication
Wikidata QID: Q125854816
full work available at URL: pmd.2012.4838
OpenAlex ID: W2078121606
review text: A domain \(\Omega\) in the complex plane is called a Denjoy domain if the boundary of \(\Omega\) is contained in the real axis. In the paper under review the authors consider the Gromov hyperbolicity of Denjoy domains equipped with the Poincaré metric and establish a necessary and sufficient condition for a Denjoy domain \(\Omega\) to be Gromov hyperbolic in terms of the symmetric fundamental domain \(\hat{\Omega}\) corresponding to \(\Omega\). Here \(\hat{\Omega}\) is a domain in the upper halfplane determined by positive numbers \(\{x_n\}\) and \(\{\rho_n\}\) related to intervals of \(\Omega \cap \mathbb{R}\) and the projection \(\pi:\mathbb{H} \to \Omega\). Also, a simple characterization of hyperbolicity is obtained, for example, a Denjoy domain with the property that \(\{x_n\}\) is unbounded and \(\lim_{n\to \infty}\rho_n/x_n\) exists is hyperbolic if and only if the limit is positive. Furthermore, comparative results are obtained, that is, for given two Denjoy domains with some property the hyperbolicity of one domain implies that of the other domain.
reviewed by: Gou Nakamura
description: scientific article; zbMATH DE number 6083232