Detours and Gromov hyperbolicity

The notion of Gromov hyperbolicity was introduced by Gromov in the setting of geometric group theory [G1], [G2], but has played an increasing role in analysis on general metric spaces [BHK], [BS], [BBo], [BBu], and extendability of Lipschitz mappings [L]. In this theory, it is often additionally assumed that the hyperbolic metric space is proper and geodesic (meaning that closed balls are compact, and each pair of points can be joined by a path whose length equals the distance between the points). These additional assumptions are useful in proofs, and valid for large classes of examples of hyperbolic spaces, for instance Cayley graphs of (finitely generated) hyperbolic groups, and certain important conformal distortions of locally compact length metrics that push the boundary of the space to infinity; see for instance [BHK, 2.8] for the case of a quasihyperbolic metric. However if the underlying metric is not locally compact, as in examples that arise in a Banach space context, then such hyperbolic conformal distortions typically fail to be proper and geodesic (although they are always length spaces). Without these added assumptions, a few “standard” results for hyperbolic spaces may fail; for one such example, see [GH, 5.13]. However, Väisälä recently proved [V1] that a large part of the theory of hyperbolic spaces goes through if we merely assume the metric is a length metric and not geodesic or proper; Väisälä then applied this theory in a Banach space context [V2]. Our paper adds to the work of [V1] by extending a characterization by Bonk of hyperbolicity in a geodesic context [B] to a length space context; see Theorem 2.1 below. Note that our version says a little more than Bonk’s result even in a geodesic space context.