We introduce the notions of asymptotic rank and injective hulls before investigating a coarse version of Dress’ 2(n+1)-inequality characterising metric spaces of combinatorial dimension at most n. This condition, referred to as (n,δ)-hyperbolicity, reduces to Gromov's quadruple definition of δ-hyperbolicity for n=1. The ℓ∞ product of n δ-hyperbolic spaces is (n,δ)-hyperbolic and, without further assumptions, any (n,δ)-hyperbolic space admits a slim (n+1)-simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. Using tools from recent developments in geometric group theory, we look at some examples related to symmetric spaces of non-compact type and Helly groups. Joint work with Urs Lang.