# Fundamental domain

of a discrete group $\Gamma$ of transformations of a topological space $X$

A subset $D \subset X$ containing elements from all the orbits (cf. Orbit) of $\Gamma$, with exactly one element from orbits in general position. There are various versions of the exact definition of a fundamental domain. Sometimes a fundamental domain is any subset belonging to a given $\sigma$- algebra (for example, the Borel $\sigma$- algebra) and containing exactly one representative from each orbit. However, if $X$ is a topological manifold, then a fundamental domain is usually a subset $D \subset X$ that is the closure of an open subset and is such that the subsets $\gamma D$, $\gamma \in \Gamma$, have pairwise no common interior points and form a locally finite covering of $X$. For example, as a fundamental domain of the group of parallel translations of the plane $\mathbf R ^ {2}$ by integer vectors one can take the square

$$\{ {( x, y) \in \mathbf R ^ {2} } : { 0 \leq x \leq 1,\ 0 \leq y \leq 1 } \} .$$

The choice of a fundamental domain is, as a rule, non-unique.

The chambers of the Weyl group $W$ are examples of fundamental domains of $W$ in its reflection representation.