  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.
