Gromov hyperbolicity of Denjoy domains through fundamental domains
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.
scientific article; zbMATH DE number 6083232