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 ${\displaystyle x\in X}$ is freely discontinuous if there exists a neighborhood U of x such that ${\displaystyle g(U)\cap U=\varnothing }$ for all ${\displaystyle g\in G}$, 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 ${\displaystyle \Omega =\Omega (G)}$. Note that ${\displaystyle \Omega }$ 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 ${\displaystyle \Omega /G}$ is a Hausdorff space.

Examples[]

The open set

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

is the free regular set of the modular group ${\displaystyle \Gamma }$ 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[]