Strong hyperbolicity
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.
Cite
@article{arxiv.1408.0250,
title = {Strong hyperbolicity},
author = {Bogdan Nica and Jan Spakula},
journal= {arXiv preprint arXiv:1408.0250},
year = {2016}
}
Comments
10 pages, new terminology, minor additions and corrections. Final version