Groups, Geometry, and Dynamics

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Volume 1, Issue 3, 2007, pp. 281–299
DOI: 10.4171/GGD/13

Published online: 2007-09-30

A characterization of hyperbolic spaces

Indira Chatterji[1] and Graham A. Niblo[2]

(1) Ohio State University, Columbus, United States
(2) University of Southampton, UK

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 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(κ) geometry for κ ≤ 0 in the same way.

Keywords: Gromov hyperbolic spaces, CAT(0) geometry, geodesic metric spaces

Chatterji Indira, Niblo Graham: A characterization of hyperbolic spaces. Groups Geom. Dyn. 1 (2007), 281-299. doi: 10.4171/GGD/13