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 | References | Similar Articles | Additional Information

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 -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:
https://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.