Properly discontinuous actions versus uniform embeddings (Q2076062)

Summary: 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.