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fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.
There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.
== 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 R''n'' of dimension ''n'', and ''G'' is the lattice Z''n'' acting on it by translations, the quotient ''X''/''G'' is the ''n''-dimensional torus. A fundamental domain ''D'' here can be taken to be [0,1)''n'', which differs from the open set (0,1)''n'' by a set of measure zero, or the closed unit cube [0,1]''n'', whose boundary consists of the points whose orbit has more than one representative in ''D''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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