Strong hyperbolicity (Q329568)

Summary: We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specically, 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 \(\mathbb H^2\). 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.