# 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 |