Uniformization, $\partial$-biLipschitz maps, sphericalization, and inversion

TL;DR

This paper introduces $ ext{partial}$-biLipschitz maps in uniform metric spaces, proves their quasim"obius nature, and explores their relation to uniformizations, sphericalization, and inversion in hyperbolic geometry.

Contribution

It establishes the equivalence between $ ext{partial}$-biLipschitz maps and biLipschitz maps in quasihyperbolic metrics, and shows uniformizations are quasim"obius equivalent, connecting uniformization, sphericalization, and inversion.

Findings

$ ext{partial}$-biLipschitz maps are quasim"obius.

Uniformizations of hyperbolic spaces are quasim"obius equivalent.

Sphericalization and inversion are compatible with uniformization.

Abstract

We define $\partial$-biLipschitz homeomorphisms between uniform metric spaces and show that these maps are always quasim\"obius. We also show that a homeomorphism being $\partial$-biLipschitz is equivalent to the map biLipschitz in the quasihyperbolic metrics on these spaces. The proofs of these claims require us to uniformize the quasihyperbolic metric. We further show that all admissible uniformizations of a Gromov hyperbolic space are quasim\"obius to one another by the identity map, including those uniformizations that are based at a point of the Gromov boundary. Using the main results we then show that the sphericalization and inversion operations are compatible with uniformization of hyperbolic spaces in a natural sense.