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

x\in X

is freely discontinuous if there exists a neighborhood U of x such that

g(U)\cap U=\emptyset

for all

g\in G

, 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

\Omega=\Omega(G)

. Note that

\Omega

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

\Omega /G

is a Hausdorff space.

Examples

The open set

\Omega(\Gamma)=\{\tau\in H: |\tau|>1 , |\tau +\overline\tau| <1\}

is the free regular set of the modular group

\Gamma

on the upper half plane H. This set is called the fundamental domain on which modular forms are studied.

Free Regular Set

Examples

See also