Groups, Geometry, and Dynamics

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Volume 10, Issue 3, 2016, pp. 951–964
DOI: 10.4171/GGD/372

Published online: 2016-09-16

Strong hyperbolicity

Bogdan Nica[1] and Ján Špakula[2]

(1) Burnside Hall, Montreal, Canada
(2) University of Southampton, UK

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.

Keywords: Hyperbolic group, Green metric, CAT(–1) space, harmonic measure

Nica Bogdan, Špakula Ján: Strong hyperbolicity. Groups Geom. Dyn. 10 (2016), 951-964. doi: 10.4171/GGD/372