properly discontinuous action

Let G be a group and E a topological spaceMathworldPlanetmath on which G acts by homeomorphismsPlanetmathPlanetmath, that is there is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ρ:GAut(E), where the latter denotes the group of self-homeomorphisms of E. The action is said to be properly discontinuous if each point eE has a neighborhoodMathworldPlanetmathPlanetmath U with the property that all non trivial elements of G move U outside itself:


For example, let p:EX be a covering map, then the group of deck transformationsMathworldPlanetmath of p acts properly discontinuously on E. Indeed if eE and DAut(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:


Assume that E is a connectedPlanetmathPlanetmath and locally path connected Hausdorff space. If the group G acts properly discontinuously on E then the quotient map p:EE/G is a covering map and Aut(p)=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