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A characterization of hyperbolic spaces

A characterization of hyperbolic spaces
A characterization of hyperbolic spaces
We show that a geodesic metric space, and in particular the Cayley graph of a finitely generated group, is hyperbolic in the sense of Gromov if and only if intersections of any two metric balls balls is itself "almost" a metric ball. In particular, R-trees are characterized among the class of geodesic metric spaces by the property that the intersection of any two metric balls is always a metric ball. A variation on the definition of "almost" allows us to characterise CAT(k) geometry for k ? 0 in the same way.
hyperbolic geometry, metric balls, bounded eccentricity, CAT(0) geometry
1661-7207
281-299
Chatterji, Indira
12ac09c3-3bf9-4d5e-b1ac-fc4ddd6b017a
Niblo, Graham A.
43fe9561-c483-4cdf-bee5-0de388b78944
Chatterji, Indira
12ac09c3-3bf9-4d5e-b1ac-fc4ddd6b017a
Niblo, Graham A.
43fe9561-c483-4cdf-bee5-0de388b78944

Chatterji, Indira and Niblo, Graham A. (2007) A characterization of hyperbolic spaces. Groups, Geometry and Dynamics, 1 (3), 281-299.

Record type: Article

Abstract

We show that a geodesic metric space, and in particular the Cayley graph of a finitely generated group, is hyperbolic in the sense of Gromov if and only if intersections of any two metric balls balls is itself "almost" a metric ball. In particular, R-trees are characterized among the class of geodesic metric spaces by the property that the intersection of any two metric balls is always a metric ball. A variation on the definition of "almost" allows us to characterise CAT(k) geometry for k ? 0 in the same way.

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Submitted date: 15 June 2006
Published date: July 2007
Keywords: hyperbolic geometry, metric balls, bounded eccentricity, CAT(0) geometry

Identifiers

Local EPrints ID: 44250
URI: https://eprints.soton.ac.uk/id/eprint/44250
ISSN: 1661-7207
PURE UUID: 1e7caadd-6082-4f76-a1c4-c8be7c7528ac

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Date deposited: 21 Feb 2007
Last modified: 13 Mar 2019 21:07

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