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 \( x\in X \) is freely discontinuous if there exists a neighborhood U of x such that \( g(U)\cap U=\varnothing \) 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.
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 Volume 287. Springer Berlin Heidelberg. pp. 15–16. ISBN 978-3-642-64878-6.
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
