Free regular set

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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.


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


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