---
tag: J5X5ED0B2Y
kind: concept
name: Properly discontinuous action
msc: [57S30, 54H15, 20H15]
specializes: [MDH7VXDCVF]
sources:
  - url: https://en.wikipedia.org/wiki/Covering_space
---

## 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|3XZRRE165N]].
Examples include lattices of translations acting on $\R^n$, Fuchsian groups acting on the hyperbolic plane, and deck groups acting on universal covers.

## Explanations

### L1 human

A group acts properly discontinuously if, from the point of view of any bounded region, only finitely many group elements move it to a nearby place.
Integer translations of the plane tile it by unit squares, and a bounded piece of the plane meets only finitely many of its shifted copies.
Gluing points that differ by a group element then gives a nice space, here a torus.
An irrational rotation of the circle is the opposite case: every arc meets infinitely many of its rotated copies, and the quotient is a bad space.