Groups, Geometry, and Dynamics

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Published online first: 2021-08-03
DOI: 10.4171/GGD/621

Properly discontinuous actions versus uniform embeddings

Kevin Schreve[1]

(1) University of Chicago, USA

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.

Keywords: Van Kampen obstruction, Wu invariant, uniformly proper dimension, action dimension

Schreve Kevin: Properly discontinuous actions versus uniform embeddings. Groups Geom. Dyn. Electronically published on August 3, 2021. doi: 10.4171/GGD/621 (to appear in print)