Group action

Properly discontinuous group action

A group acting on a topological space is called properly discontinuous¹ iff every has a neighbourhood such that for every with , . topology

Properties

1. A properly discontinuous group action is necessarily free.
2. If is also topological group and acts continuously, then the orbit map is a homeomorphism of discrete topological spaces.

Proof of properties 1–2

Let be a topological group acting on continuously and properly discontinuously.

Assume that does not act freely, i.e. there exist with such that for some . Then for any neighbourhood of , , violating proper discontinuity. Thus acts freely.

Now consider the orbit of a point with its subspace topology and the corresponding orbit map .

Assume there exists with not open in . Let be an open neighbourhood of in . Since is a homeomorphism, is open in , and thus is open in , so at least one distinct point is contained in . Then , violating proper discontinuity. Therefore must be discrete.

Now clearly the orbit map is continuous and bijective (injectivity by freeness, surjectivity by construction). Thus every singleton in is the preïmage of a singleton in and is therefore open. Therefore is discrete, and is a homeomorphism, since the inverse is continuous as a map between discrete spaces.

3. Orbit space of a properly discontinuous group action covers .

---

tidy | en | sembr

Footnotes

1. German eigentlich diskontinuierlich ↩