# Properly discontinuous actions versus uniform embeddings

## TL;DR

This paper investigates the distinction between proper group actions and uniform embeddings into contractible manifolds, revealing limitations of group actions on higher-dimensional manifolds based on product constructions.

## Contribution

It demonstrates that certain product groups, which can uniformly embed into a manifold, cannot act properly on a higher-dimensional contractible manifold.

## Key findings

- Certain product groups do not act properly on higher-dimensional contractible manifolds.
- Examples of groups that embed uniformly but do not act properly are extended to products.
- The results highlight differences between proper actions and uniform embeddings in geometric group theory.

## Abstract

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.