# properly discontinuous action

Let $G$ be a group and $E$ a topological space on which $G$ acts by homeomorphisms, that is there is a homomorphism $\rho\colon\thinspace G\to\operatorname{Aut}(E)$, where the latter denotes the group of self-homeomorphisms of $E$. The action is said to be properly discontinuous if each point $e\in E$ has a neighborhood $U$ with the property that all non trivial elements of $G$ move $U$ outside itself:

 $\forall g\in G\quad g\neq\operatorname{id}\Rightarrow gU\cap U=\emptyset\,.$

For example, let $p\colon\thinspace E\to X$ be a covering map, then the group of deck transformations of $p$ acts properly discontinuously on $E$. Indeed if $e\in E$ and $D\in\operatorname{Aut}(p)$ then one can take as $U$ to be any neighborhood with the property that $p(U)$ is evenly covered. The following shows that this is the only example:

###### Theorem.

Assume that $E$ is a connected and locally path connected Hausdorff space. If the group $G$ acts properly discontinuously on $E$ then the quotient map $p\colon\thinspace E\to E/G$ is a covering map and $\operatorname{Aut}(p)=G$.

Title properly discontinuous action ProperlyDiscontinuousAction 2013-03-22 13:28:11 2013-03-22 13:28:11 Dr_Absentius (537) Dr_Absentius (537) 9 Dr_Absentius (537) Definition msc 55R05 msc 37B05 DiscontinuousAction DeckTransformation properly discontinuous properly discontinuously