  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) $\rho: G \rightarrow X$ 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 only virtually act on a contractible
$3$-manifold. We show that $k$-fold products of these examples do not act on a
contractible $3k$-manifold.
