A lower bound to the action dimension of a group

Abstract

The action dimension of a discrete group $\Gamma$, $actdim\left(\Gamma \right)$, is defined to be the smallest integer $m$ such that $\Gamma$ admits a properly discontinuous action on a contractible $m$–manifold. If no such $m$ exists, we define $actdim\left(\Gamma \right)\equiv \infty$. Bestvina, Kapovich, and Kleiner used Van Kampen’s theory of embedding obstruction to provide a lower bound to the action dimension of a group. In this article, another lower bound to the action dimension of a group is obtained by extending their work, and the action dimensions of the fundamental groups of certain manifolds are found by computing this new lower bound.