A characterization of hyperbolic spaces

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 is itself “almost” a metric ball. In particular, $\mathbb{R}$-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 $\mathrm{CAT}(\kappa )$ geometry for $\kappa \le 0$ in the same way.