Uniformizing Gromov hyperbolic spaces with Busemann functions

TL;DR

This paper introduces a method to transform Gromov hyperbolic spaces into uniform metric spaces using Busemann functions, enabling boundary identification and optimal uniformization in certain spaces.

Contribution

It generalizes the uniformization process for hyperbolic spaces, linking Gromov boundaries with uniformized spaces and providing optimal procedures for CAT(-1) spaces.

Findings

Constructed an incomplete uniform space from Gromov hyperbolic space using Busemann functions.

Established a biLipschitz boundary identification between the uniformized space and the original boundary.

Demonstrated optimal uniformization exponents for CAT(-1) spaces.

Abstract

Given a complete Gromov hyperbolic space $X$ that is roughly starlike from a point $\omega$ in its Gromov boundary $\partial_{G}X$, we use a Busemann function based at $\omega$ to construct an incomplete unbounded uniform metric space $X_{\varepsilon}$ whose boundary $\partial X_{\varepsilon}$ can be canonically identified with the Gromov boundary $\partial_{\omega}X$ of $X$ relative to $\omega$. This uniformization construction generalizes the procedure used to obtain the Euclidean upper half plane from the hyperbolic plane. Furthermore we show, for an arbitrary metric space $Z$, that there is a hyperbolic filling $X$ of $Z$ that can be uniformized in such a way that the boundary $\partial X_{\varepsilon}$ has a biLipschitz identification with the completion $\bar{Z}$ of $Z$. We also prove that this uniformization procedure can be done at an exponent that is often optimal in the case of CAT$(-1)$ spaces.