Uniformization of intrinsic Gromov hyperbolic spaces

TL;DR

This paper develops a uniformization method for Gromov hyperbolic spaces, transforming them into bounded uniform spaces via conformal deformation, and establishes a natural quasi-isometry between their boundaries.

Contribution

It generalizes previous results by providing a uniformization procedure applicable to non-geodesic, non-proper Gromov hyperbolic spaces, linking their boundaries through quasi-isometry.

Findings

Conformal deformation yields bounded uniform spaces from Gromov hyperbolic spaces.

Establishes a natural quasi-isometry between Gromov boundary and metric boundary.

Generalizes prior uniformization results to broader classes of hyperbolic spaces.

Abstract

The purpose of this paper is to provide a uniformization procedure for Gromov hyperbolic spaces, which need not be geodesic or proper. We prove that the conformal deformation of a Gromov hyperbolic space is a bounded uniform space. Further, we show that there is a natural quasi-isometry between the Gromov boundary and the metric boundary of the deformed space. Our main results are a generalization of the results of Bonk, Heninonen, and Koskela [Proposition 4.5, Proposition 4.13, Ast\'{e}risque 270 (2001)].