Free regular set

In mathematics, a free regular set is a subset of a topological space that is acted upon disjointly under a given group action. To be more precise, let X be a topological space. Let G be a group of homeomorphisms from X to X. Then we say that the action of the group G at a point [math]\displaystyle{ x\in X }[/math] is freely discontinuous if there exists a neighborhood U of x such that [math]\displaystyle{ g(U)\cap U=\varnothing }[/math] for all [math]\displaystyle{ g\in G }[/math], excluding the identity. Such a U is sometimes called a nice neighborhood of x.

The set of points at which G is freely discontinuous is called the free regular set and is sometimes denoted by [math]\displaystyle{ \Omega=\Omega(G) }[/math]. Note that [math]\displaystyle{ \Omega }[/math] is an open set.

If Y is a subset of X, then Y/G is the space of equivalence classes, and it inherits the canonical topology from Y; that is, the projection from Y to Y/G is continuous and open.

Note that [math]\displaystyle{ \Omega /G }[/math] is a Hausdorff space.

Examples

The open set

[math]\displaystyle{ \Omega(\Gamma)=\{\tau\in H: |\tau|\gt 1 , |\tau +\overline\tau| \lt 1\} }[/math]

is the free regular set of the modular group [math]\displaystyle{ \Gamma }[/math] on the upper half-plane H. This set is called the fundamental domain on which modular forms are studied.

See also

• Covering map
• Klein geometry
• Homogeneous space
• Clifford–Klein form
• G-torsor