Strong hyperbolicity

TL;DR

This paper introduces the concept of strong hyperbolicity, a refined form of hyperbolic spaces with enhanced boundary properties, demonstrating its applicability to CAT(-1) spaces, hyperbolic planes, and random walks on hyperbolic groups.

Contribution

It defines strong hyperbolicity, proves CAT(-1) spaces are strongly hyperbolic, and shows the Green metric from random walks is strongly hyperbolic, linking geometric and probabilistic properties.

Findings

CAT(-1) spaces are strongly hyperbolic

The hyperbolic plane's hyperbolicity constant is determined

Green metric from random walks is strongly hyperbolic

Abstract

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(-1) spaces are strongly hyperbolic. On the way, we determine the best constant of hyperbolicity for the standard hyperbolic plane. We also show that the Green metric defined by a random walk on a hyperbolic group is strongly hyperbolic. A measure-theoretic consequence at the boundary is that the harmonic measure defined by a random walk is a visual Hausdorff measure.