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 $\rho :G\to \mathrm{Aut}(E)$, 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:

$$\forall g\in G\mathit{\hspace{1em}}g\ne \mathrm{id}\Rightarrow gU\cap U=\mathrm{\varnothing }.$$

For example, let $p:E\to X$ be a covering map, then the group of deck transformations of $p$ acts properly discontinuously on $E$. Indeed if $e\in E$ and $D\in \mathrm{Aut}(p)$ 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 $p\mathrm{:}E\mathrm{\to }E\mathrm{/}G$ is a covering map and $\mathrm{Aut}\mathit{}\mathrm{(}p\mathrm{)}\mathrm{=}G$.

Title	properly discontinuous action
Canonical name	ProperlyDiscontinuousAction
Date of creation	2013-03-22 13:28:11
Last modified on	2013-03-22 13:28:11
Owner	Dr_Absentius (537)
Last modified by	Dr_Absentius (537)
Numerical id	9
Author	Dr_Absentius (537)
Entry type	Definition
Classification	msc 55R05
Classification	msc 37B05
Related topic	DiscontinuousAction
Related topic	DeckTransformation
Defines	properly discontinuous
Defines	properly discontinuously