Rough isometry between Gromov hyperbolic spaces and uniformization

Abstract

In this note we show that given two complete geodesic Gromov hyperbolic spaces that are roughly isometric and an arbitrary $ϵ>0$ (not necessarily small), either the uniformization of both spaces with parameter $ϵ$ results in uniform domains, or else neither uniformized space is a uniform domain. The terminology of "uniformization" is from [BHK], where it is shown that the uniformization, with parameter $ϵ>0$, of a complete geodesic Gromov hyperbolic space results in a uniform domain provided $ϵ$ is small enough.