# 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