We know that if we want to construct a space with a given fundamental group $G$ ,we can use cells and attaching maps, or fundamental domains and attaching maps, as in : How to determine space with a given fundamental group.
But there is a different way: if $M$ is, I think a manifold, and $G$ acts proper-discontinuously on $M$, then the quotient $ M/G$ has $G$ as a fundamental group. EDIT: As Seirios points out, $M$ may have to be simply-connected.
Now, can we always do this for any group $G$, i.e., can we always construct a space with fundamental group $G$ by defining a proper discontinuous action $M/G$? Can we always find $M$ so that the action is of the necessary type? Thanks.