Van Kampen's embedding obstruction for discrete groups

TL;DR

This paper establishes a lower bound on the dimension of contractible manifolds suitable for proper group actions, demonstrating that certain free group products cannot act properly on specific Euclidean spaces.

Contribution

It introduces a new lower bound for the dimension of manifolds allowing proper discontinuous actions of groups, highlighting restrictions for free group products.

Findings

Nonabelian free groups' n-fold product cannot act properly on R^{2n-1}

Provides a lower bound for the dimension of contractible manifolds for group actions

Shows limitations of group actions in Euclidean spaces

Abstract

We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the $n$-fold product of nonabelian free groups cannot act properly discontinuously on $\R^{2n-1}$.