Isoperimetric Inequalities and the Asymptotic Rank of Metric Spaces

In this article we study connections between the asymptotic rank of a metric space and higher-dimensional isoperimetric inequalities. We work in the class of metric spaces admitting cone type inequalities which, in particular, includes all Hadamard spaces, i. e. simply connected metric spaces of nonpositive curvature in the sense of Alexandrov. As was shown by Gromov, spaces with cone type inequalities admit isoperimetric inequalities of at most Euclidean type. Here we prove that they admit isoperimetric inequalities of sub-Euclidean type for k-cycles whenever k is greater or equal to their asymptotic rank. As a consequence it follows that the higher-dimensional isoperimetric inequalities can be used to detect the asymptotic rank of such spaces. Our work is to some extent inspired by a conjecture of Gromov which, in the case of proper cocompact Hadamard spaces, asserts even linear isoperimetric inequalities above the asymptotic rank. Our methods can moreover be used to establish polynomial isoperimetric inequalities for metric spaces admitting polynomial cone type inequalities. These include spaces with polynomial Lipschitz combings.