nLab properly discontinuous action

Definition

The action of a topological group $G$ on a topological space $X$ is called properly discontinuous if every point $x \in X$ has a neighbourhood $U_x$ such that the intersection $g(U_x) \cap U_x$ with its translate under the group action via some element $g \in G$ is non-empty only for the neutral element $e \in G$:

$$g(U_x) \cap U_x \neq \emptyset \phantom{AA} \Rightarrow \phantom{AA} g = e$$

This is equivalent to the condition that the quotient space coprojection $X \longrightarrow X/G$ is a covering space-projection.

Therefore properly discontinuous actions are also called covering space actions (Hatcher).

Related concepts

• proper action

• transitive action, free action, semi-free action, regular action

• Clifford-Klein form

References

• Jack Lee, chapter 12 of Introduction to Topological Manifolds

• Jack Lee, chapter 21 of Introduction to Smooth Manifolds

• Jack Lee, MO comment, Dec. 2014

• Alan Hatcher, Algebraic Topology