Properly discontinuous actions versus uniform embeddings

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