Metric transforms yielding Gromov hyperbolic spaces

TL;DR

This paper characterizes metric transforms that preserve Gromov hyperbolicity in spaces, especially the Euclidean half line, and establishes rigidity results for roughly geodesic hyperbolic spaces under such transforms.

Contribution

It provides a complete characterization of approximately nondecreasing, unbounded metric transforms that yield Gromov hyperbolic spaces, and proves a rigidity result for these transforms in roughly geodesic hyperbolic spaces.

Findings

Characterization of metric transforms preserving Gromov hyperbolicity.

Identification of conditions under which the Euclidean half line remains hyperbolic.

Rigidity of hyperbolic spaces under metric transforms that are approximate dilations.

Abstract

A real valued function $\varphi$ of one variable is called a metric transform if for every metric space $(X,d)$ the composition $d_\varphi = \varphi\circ d$ is also a metric on $X$. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms $\varphi$ such that the transformed Euclidean half line $([0,\infty),|\cdot|_\varphi)$ 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 $\varphi$ is an approximately nondecreasing, unbounded metric transform such that the transformed space $(X,d_\varphi)$ is Gromov hyperbolic and roughly geodesic then $\varphi$ is an approximate dilation and the original space $(X,d)$ is Gromov hyperbolic and roughly geodesic.