A very simple characterization of Gromov hyperbolicity for a special kind of Denjoy domains

Ana Portilla; Jos\'e M. Rodr{\'\i}guez; Eva Tour{\'\i}s

doi:10.4134/JKMS.2011.48.3.565

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2011; 48(3): 565-583

Printed May 1, 2011

https://doi.org/10.4134/JKMS.2011.48.3.565

A very simple characterization of Gromov hyperbolicity for a special kind of Denjoy domains

Ana Portilla, Jos\'e M. Rodr{\'\i}guez, Eva Tour{\'\i}s

St. Louis University, Universidad Carlos III de Madrid, Universidad Carlos III de Madrid

Abstract

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Abstract

In this paper we provide characterizations for the Gromov hyperbolicity of some particular Denjoy domains and besides some sufficient conditions to guarantee or discard the hyperbolicity of some others. The conditions obtained involve just the lengths of some special simple closed geodesics in the domain. These results, on the one hand, show the extraordinary complexity of determining the hyperbolicity of a domain and, on the other hand, allow us to construct easily a large variety of both hyperbolic and non-hyperbolic domains.

Keywords: Denjoy domain, Gromov hyperbolicity, Poincar\'e metric, train

MSC numbers: Primary 30F45; Secondary 53C23, 30C99