Properly discontinuous actions versus uniform embeddings

Whenever a finitely generated group G acts properly discontinuously by isometries on a metric space X , there is an induced uniform embedding (a Lipschitz and uniformly proper map) \rho\colon G \rightarrow X given by mapping G to an orbit. We study when there is a difference between a finitely generated group G acting properly on a contractible n -manifold and uniformly embedding into a contractible n -manifold. For example, Kapovich and Kleiner showed that there are torsion-free hyperbolic groups that uniformly embed into a contractible 3 -manifold but do not act on a contractible 3 -manifold. We show that k -fold products of certain examples do not act on contractible 3k -manifolds.