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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Gromov hyperbolicity of the $j_G$ and ${\tilde\jmath}_G$ metrics


Author: Peter A. Hästö
Journal: Proc. Amer. Math. Soc. 134 (2006), 1137-1142
MSC (2000): Primary 30F45; Secondary 53C23, 30C99
Published electronically: August 29, 2005
MathSciNet review: 2196049
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Abstract: In this note it is shown that the ${\tilde\jmath}_G$ metric is always Gromov hyperbolic, but that the $j_G$ metric is Gromov hyperbolic if and only if $G$ has exactly one boundary point. As a corollary we get a new proof for the fact that the quasihyperbolic metric is Gromov hyperbolic in uniform domains.


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Additional Information

Peter A. Hästö
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
Address at time of publication: Department of Mathematics and Statistics, P.O. Box 68, FIN-00014 University of Helsinki, Finland
Email: peter.hasto@helsinki.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08053-6
PII: S 0002-9939(05)08053-6
Keywords: Gromov hyperbolic, $j_G$ metric, ${\tilde\jmath}_G$ metric, quasihyperbolic metric
Received by editor(s): March 3, 2004
Received by editor(s) in revised form: November 3, 2004
Published electronically: August 29, 2005
Additional Notes: The author was supported in part by a Gehring-Finland Post-doctoral Fellowship and by the Finnish Academy of Science and Letters.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.