Some remarks on the Gehring-Hayman theorem

TL;DR

This paper offers new characterizations of the Gehring-Hayman theorem using Gromov boundary and uniformity concepts, and identifies critical exponents for uniformity in hyperbolic and model spaces.

Contribution

It introduces novel characterizations of the Gehring-Hayman theorem and determines critical exponents for uniformity in various hyperbolic and model spaces.

Findings

New characterizations of the Gehring-Hayman theorem via Gromov boundary and uniformity

Identification of critical exponents for uniformity in hyperbolic spaces and model spaces

Extension of results to hyperbolic fillings and spaces with negative curvature

Abstract

In this paper we provide new characterizations of the Gehring-Hayman theorem from the point of view of Gromov boundary and uniformity. We also determine the critical exponents for the uniformized space to be a uniform space in the case of the hyperbolic spaces, the model spaces $\mathbb{M}^{\kappa}_n$ of the sectional curvature $\kappa<0$ with the dimension $n \geq 2$ and hyperbolic fillings.