Quasi-isometries and the de Rham decomposition

We study quasi-isometries Φ:∏Xi→∏Yj of product spaces and find conditions on the Xi, Yj which guarantee that the product structure is preserved. The main result applies to universal covers of compact Riemannian manifolds with nonpositive sectional curvature. We introduce a quasi-isometry invariant notion of coarse rank for metric spaces which coincides with the geometric rank for universal covers of closed nonpositively curved manifolds. This shows that the geometric rank is a quasi-isometry invariant.