Properly discontinuous action

In topology and related branches of mathematics, an action of a group G on a topological space X is called proper if the map from G×X to X×X taking (g,x) to (gx,x) is proper, and is called properly discontinuous if in addition G is discrete. There are several other similar but inequivalent properties of group actions that are often confused with properly discontinuous actions.

Properly discontinuous action

A (continuous) group action of a topological group G on a topological space X is called proper if the map from G×X to X×X taking (g,x) to (gx,x) is proper. If in addition the group G is discrete then the action is called properly discontinuous (tom Dieck 1987, p. 29).

Equivalently, an action of a discrete group G on a topological space X is properly discontinuous if and only if any two points x and y have neighborhoods U_x and U_y such that there are only a finite number of group elements g with g(U_x) meeting U_y.

In the case of a discrete group G acting on a locally compact Hausdorff space X, an equivalent definition is that the action is called properly discontinuous if for all compact subsets K of X there are only a finite number of group elements g such that K and g(K) meet.

A key property of properly discontinuous actions is that the quotient space X/G is Hausdorff.

Example

Suppose that H is a locally compact Hausdorff group with a compact subgroup K. Then H acts on the quotient space X=H/K. A subgroup G of H acts properly discontinuously on X if and only if G is a discrete subgroup of H.

Similar properties

There are several other properties of group actions that are not equivalent to proper discontinuity but are frequently confused with it.

Wandering actions

A group action is called wandering or sometimes discontinuous if every point x of X has a neighborhood U that meets gU for only a finite number of elements g of G.

If X is the plane with the origin missing, and G is the infinite cyclic group generated by (x,y)→(2x,y/2) then this action is wandering but not properly discontinuous, and the quotient space is non-Hausdorff. The problem is that any neighborhood of (1,0) has infinitely many conjugates that intersect any given neighborhood of (0,1).

Discrete orbits

The group action has discrete orbits and is sometimes called discontinuous if for any two points x, y there is a neighborhood of y containing gx for only a finite number of g in G. This is equivalent to saying that the stabilizers of points are finite and every orbit has empty limit set (Thurston 1980).

See also

• Homogeneous space
• G-torsor
• Wandering set

References

• tom Dieck, Tammo (1987), Transformation groups, de Gruyter Studies in Mathematics, 8, Berlin: Walter de Gruyter & Co., ISBN 978-3-11-009745-0, MR889050, http://books.google.com/books?id=azcQhi6XeioC 
• Thurston, William (1980), The geometry and topology of three-manifolds, Princeton lecture notes, http://library.msri.org/books/gt3m/