A combinatorial higher-rank hyperbolicity condition

We investigate a coarse version of a 2 (n+1) -point inequality characterizing metric spaces of combinatorial dimension at most n due to Dress. This condition, experimentally called (n, ) -hyperbolicity, reduces to Gromov’s quadruple definition of -hyperbolicity in case n = 1. The l_ -product of n -hyperbolic spaces is (n, ) -hyperbolic. Every (n, ) -hyperbolic metric space, without any further assumptions, possesses a slim (n+1) -simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. In connection with recent work in geometric group theory, we show that every Helly group and every hierarchically hyperbolic group of (asymptotic) rank n acts geometrically on some (n, ) -hyperbolic space.