The action dimension of a discrete group , , is defined to be the smallest integer such that admits a properly discontinuous action on a contractible –manifold. If no such exists, we define . 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.
"A lower bound to the action dimension of a group." Algebr. Geom. Topol. 4 (1) 273 - 296, 2004. https://doi.org/10.2140/agt.2004.4.273