# 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.^{[1]}
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

## References

- ↑ Maskit, Bernard (1987).
*Discontinuous Groups in the Plane*. Grundlehren der mathematischen Wissenschaften.**287**. Springer. pp. 15–16. ISBN 978-3-642-64878-6.

Original source: https://en.wikipedia.org/wiki/Free regular set.
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