Uniformizing Gromov hyperbolic spaces with Busemann functions

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 $X_{\epsilon }$ whose boundary $\partial X_{\epsilon }$ 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 $\partial X_{\epsilon }$ 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.

48 pages. v4: Extensive revisions. New theorem added on uniformizing CAT(-1) spaces

References in corpus (3)

• Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
• Extension and trace theorems for noncompact doubling spaces
• Doubling and Poincaré inequalities for uniformized measures on Gromov hyperbolic spaces

Cited by in corpus (5)

• Extension and trace results for doubling metric measure spaces and their hyperbolic fillings
• Extension and trace theorems for noncompact doubling spaces
• Doubling and Poincaré inequalities for uniformized measures on Gromov hyperbolic spaces
• Uniformization, $\partial$-biLipschitz maps, sphericalization, and inversion
• Gromov-Hausdorff distance with boundary and its application to Gromov hyperbolic spaces and uniform spaces