Strong hyperbolicity
Bogdan Nica
Burnside Hall, Montreal, CanadaJán Špakula
University of Southampton, UK
![Strong hyperbolicity cover](/_next/image?url=https%3A%2F%2Fcontent.preview.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-ggd-volume-10-issue-3.png&w=3840&q=90)
Abstract
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 . 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 this article
Bogdan Nica, Ján Špakula, Strong hyperbolicity. Groups Geom. Dyn. 10 (2016), no. 3, pp. 951–964
DOI 10.4171/GGD/372