A properly discontinuous group action produces a normal cover whose deck group is the acting group.Suggest a slogan
Let a group \(G\) act on a space \(Y\) such that each \(y\in Y\) has a neighborhood \(U\) with \(g_1U\cap g_2U = \emptyset\) whenever \(g_1\neq g_2\) in \(G\). Then
the quotient map \(p\colon Y \to Y/G\), \(p(y) = Gy\), is a normal covering space;
if \(Y\) is path-connected, then \(G\) is the group of deck transformations of \(p\);
if \(Y\) is path-connected and locally path-connected, then \(G\cong \pi_1(Y/G) / p_*(\pi_1(Y))\)
[@Hat02].