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

   

 

Hyperbolizing metric spaces


Author: Zair Ibragimov
Journal: Proc. Amer. Math. Soc. 139 (2011), 4401-4407
MSC (2010): Primary 30F45; Secondary 53C23, 30C99
Published electronically: April 25, 2011
MathSciNet review: 2823085
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Abstract: It was proved by M. Bonk, J. Heinonen and P. Koskela that the quasihyperbolic metric hyperbolizes (in the sense of Gromov) uniform metric spaces. In this paper we introduce a new metric that hyperbolizes all locally compact noncomplete metric spaces. The metric is generic in the sense that (1) it can be defined on any metric space; (2) it preserves the quasiconformal geometry of the space; (3) it generalizes the $ j$-metric, the hyperbolic cone metric and the hyperbolic metric of hyperspaces; and (4) it is quasi-isometric to the quasihyperbolic metric of uniform metric spaces. In particular, the Gromov hyperbolicity of these metrics also follows from that of our metric.


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Additional Information

Zair Ibragimov
Affiliation: Department of Mathematics, California State University, Fullerton, California 92831
Email: zibragimov@fullerton.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10857-8
Keywords: Metric spaces, Gromov hyperbolicity, quasihyperbolic metric
Received by editor(s): September 9, 2010
Received by editor(s) in revised form: October 20, 2010
Published electronically: April 25, 2011
Dedicated: Dedicated to Fred Gehring on the occasion of his 85th birthday
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.