  In this paper we study a homological version of the higher-dimensional
divergence invariants defined by Brady and Farb. We show that they are
quasi-isometry invariants in the class of proper cocompact Hadamard spaces in
the sense of Alexandrov and that they can moreover be used to detect the
Euclidean rank of such spaces. We thereby extend results of Brady-Farb,
Leuzinger, and Hindawi from the setting of symmetric spaces of non-compact type
to that of singular Hadamard spaces. Finally, we exhibit the optimal power for
the growth of the divergence above the rank for symmetric spaces of non-compact
type and for Hadamard spaces with strictly negative upper curvature bound.
