Fundamental domain

Fundamental domain

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 Zⁿ, 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 SL₂(R), which has dimension 3, but the other dimension is accounted for by a U(1) group which being compact is nothing serious).