Fundamental Domain - Hints at General Definition

Hints At General Definition

Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain (also called fundamental region) for this action is a set D of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several precisely defined ways. One typical condition is that D is almost an open set, in the sense that D is the symmetric difference of an open set in G with a set of measure zero, for a certain (quasi)invariant measure on X. A fundamental domain always contains a free regular set U, an open set moved around by G into disjoint copies, and nearly as good as D in representing the orbits. Frequently D is required to be a complete set of coset representatives with some repetitions, but the repeated part has measure zero. This is a typical situation in ergodic theory. If a fundamental domain is used to calculate an integral on X/G, sets of measure zero do not matter.

For example, when X is Euclidean space Rn of dimension n, and G is the lattice Zn acting on it by translations, the quotient X/G is the n-dimensional torus. A fundamental domain D here can be taken to be n, whose boundary consists of the points whose orbit has more than one representative in D.