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.

Comments

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