Strong hyperbolicity

  • Bogdan Nica

    Burnside Hall, Montreal, Canada
  • Ján Špakula

    University of Southampton, UK

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 H2\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.

Cite this article

Bogdan Nica, Ján Špakula, Strong hyperbolicity. Groups Geom. Dyn. 10 (2016), no. 3, pp. 951–964

DOI 10.4171/GGD/372