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.
Cite this article
Indira Chatterji, Graham A. Niblo, A characterization of hyperbolic spaces. Groups Geom. Dyn. 1 (2007), no. 3, pp. 281–299DOI 10.4171/GGD/13