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Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains
Author(s) -
Hästö Peter,
Portilla Ana,
Rodríguez José M.,
Tourís Eva
Publication year - 2010
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp125
Subject(s) - mathematics , equivalence (formal languages) , hyperbolic group , pure mathematics , euclidean geometry , hyperbolic 3 manifold , metric (unit) , hyperbolic manifold , relatively hyperbolic group , hyperbolic space , metric space , domain (mathematical analysis) , hyperbolic function , mathematical analysis , geometry , operations management , economics
In this article we investigate the Gromov hyperbolicity of Denjoy domains equipped with the hyperbolic or the quasihyperbolic metric. We first prove the existence of suitable families of quasigeodesics. The main result shows that a Denjoy domain is Gromov hyperbolic with respect to the hyperbolic metric if and only it is Gromov hyperbolic with respect to the quasihyperbolic metric. Using these tools we give a characterization in terms of Euclidean distances of when the domains are Gromov hyperbolic. We also give several concrete examples of families of domains satisfying the criteria of the theorems.