Gromov Hyperbolic Equivalence of the Hyperbolic and Quasihyperbolic Metrics in Denjoy Domains Articles uri icon

publication date

  • February 2010

start page

  • 282

end page

  • 294

volume

  • 42

international standard serial number (ISSN)

  • 0024-6093

electronic international standard serial number (EISSN)

  • 1469-2120

abstract

  • 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.