Properly discontinuous action

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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[edit]

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 of X have neighborhoods Ux and Uy such that there are only a finite number of group elements g with g(Ux) meeting Uy.

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[edit]

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[edit]

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

Wandering actions[edit]

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[edit]

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[edit]

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