In mathematics, given a lattice Γ in a Lie group *G*, a **fundamental domain** is a set *D* of representatives for the cosets *G*/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several 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 0, for the Haar measure on *G*. For example, when *G* is Euclidean space of dimension *n*, and Γ is **Z**^{n}, the quotient *G*/Γ is the *n*-torus. A fundamental domain (also called **fundamental region**) here can be taken to be [0,1)^{n}, which is the open set (0,1)^{n} up to a set of measure zero. In practice the main use of a fundamental domain may be to compute integrals on *G*/Γ, in which case the set of measure zero is mentioned only to keep straight the pedantic assertion that *D* is *exactly* a set of coset representatives, and may quickly be forgotten. Other uses, for example in ergodic theory, are similarly based on having a reasonable set *D* up to sets of measure zero. The existence and description of a fundamental domain is in general something requiring painstaking work to establish. For the case of the modular group, there is a famous diagram appearing in all classical books on elliptic modular functions, showing a set in the upper half plane that is the basis for the construction of a fundamental domain (in this case the modular group is given as a subgroup of *SL*_{2}(**R**), which has dimension 3, but the other dimension is accounted for by a U(1) group which being compact is nothing serious). In other usages, a **fundamental domain** is simply required to map finite-to-one in the quotient. See also: Brillouin zone |