Properly discontinuous action

Summary

An action of a group by homeomorphisms is properly discontinuous when each compact set meets only finitely many of its translates. On a locally compact Hausdorff space, it is the same as a proper action of the group with the discrete topology. A free properly discontinuous action of a group on a manifold gives a quotient manifold, and the quotient map is a covering map. Examples include lattices of translations acting on $\R^n$, Fuchsian groups acting on the hyperbolic plane, and deck groups acting on universal covers.