A free properly discontinuous action on a simply connected space realizes the acting group as the quotient fundamental group.Suggest a slogan
Let \(G\) act on a simply connected, locally path-connected space \(Y\) such that every \(y\in Y\) has a neighborhood \(U\) with \(gU\cap U = \emptyset\) for all \(g\neq 1\) in \(G\). Then \(Y\to Y/G\) is a universal cover and \(\pi_1(Y/G)\cong G\) [@Hat02].