In topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 and related branches of mathematics, an action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 of a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 G on a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 X is called proper if the map from G×X to X×X taking (g,x) to (gx,x) is proper

Proper map

In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.- Definition :...

, 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 .

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 .