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August 28, 2019

Proper Actions versus Uniform Embeddings

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) $f \colon G \to X$ given by mapping $G$ to an orbit. I will talk about some examples of groups which uniformly embed into a contractible n-manifold but do not act on a contractible n-manifold. Kapovich and Kleiner constructed torsion-free hyperbolic groups that embed into $\mathbb{R}^3$ but only act on $\mathbb{R}^4$. My main application will be showing that certain k-fold direct products of these groups do not act on $\mathbb{R}^{3k}$.