Metric transforms yielding Gromov hyperbolic spaces

Abstract

A real valued function ϕ of one variable is called a metric transform if for every metric space (X, d) the composition dϕ = ϕ • d is also a metric on X. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms ϕ such that the transformed Euclidean half line ([0, ∞), | · |ϕ) is Gromov hyperbolic. As a consequence, we obtain metric transform rigidity for roughly geodesic Gromov hyperbolic spaces, that is, if (X, d) is any metric space containing a rough geodesic ray and ϕ is an approximately nondecreasing, unbounded metric transform such that the transformed space (X, dϕ) is Gromov hyperbolic and roughly geodesic then ϕ is an approximate dilation and the original space (X, d) is Gromov hyperbolic and roughly geodesic.