The asymptotic rank of metric spaces

In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher isoperimetric filling functions. For a proper, cocompact, simply connected geodesic metric space of non-positive curvature in the sense of Alexandrov the asymptotic rank equals its Euclidean rank.