Quasihyperbolic geodesics are hyperbolic quasi-geodesics

TL;DR

This paper proves that in hyperbolic plane domains, hyperbolic and quasihyperbolic quasi-geodesics coincide and demonstrates the Gromov hyperbolicity of these domains under both metrics, enhancing understanding of their large-scale geometry.

Contribution

It establishes the equivalence of hyperbolic and quasihyperbolic quasi-geodesics and shows the Gromov hyperbolicity of hyperbolic plane domains with these metrics.

Findings

Hyperbolic and quasihyperbolic quasi-geodesics are the same curves in hyperbolic plane domains.

Hyperbolic plane domains are Gromov hyperbolic under both metrics.

The large-scale geometry of these domains is characterized by Gromov hyperbolicity.

Abstract

This is a tale describing the large scale geometry of Euclidean plane domains with their hyperbolic or quasihyperbolic distances. We prove that in any hyperbolic plane domain, hyperbolic and quasihyperbolic quasi-geodesics are the same curves. We also demonstrate the simultaneous Gromov hyperbolicity of such domains with their hyperbolic or quasihyperbolic distances.