Whenever a finitely generated group acts properly discontinuously by isometries on a metric space , there is an induced uniform embedding (a Lipschitz and uniformly proper map) given by mapping to an orbit. We study when there is a difference between a finitely generated group acting properly on a contractible -manifold and uniformly embedding into a contractible -manifold. For example, Kapovich and Kleiner showed that there are torsion-free hyperbolic groups that uniformly embed into a contractible -manifold but do not act on a contractible -manifold. We show that -fold products of certain examples do not act on contractible -manifolds.
Cite this article
Kevin Schreve, Properly discontinuous actions versus uniform embeddings. Groups Geom. Dyn. 15 (2021), no. 3, pp. 1015–1039DOI 10.4171/GGD/621