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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. 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 cosets. 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 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.

In other usages, a fundamental domain is simply required to map finite-to-one in the quotient.

Example

The existence and description of a fundamental domain is in general something requiring painstaking work to establish. The diagram to the right shows part of the construction of the fundamental domain for the action of the modular group Γ on the upper half plane H.

This famous diagram appears in all classical books on elliptic modular functions. (It was probably well known to C. F. Gauss, who dealt with fundamental domains in the guise of the reduction theory of quadratic forms .) Here, each triangular region (bounded by the blue lines) is a free regular set of the action of Γ on H. The boundaries (the blue lines) are not a part of the free regular sets. To construct a fundamental domain of H/Γ, one must also consider how to assign points on the boundary, being careful not to double-count such points. Thus, the free regular set in this example is

$U = \left\{ z \in H: \left| z \right| > 1,\, \left| \,\mbox{Re}(z) \,\right| < \frac{1}{2} \right\}.$

The fundamental domain is built by adding the boundary on the left plus half the arc on the bottom:

$D=U\cup\left\{ z \in H: \left| z \right| \geq 1,\, \mbox{Re}(z)=\frac{-1}{2} \right\} \cup \left\{ z \in H: \left| z \right| = 1,\, \mbox{Re}(z)<0 \right\}.$

The choice of which points of the boundary to include as a part of the fundamental domain is arbitrary, and varies from author to author.

The core difficulty of defining the fundamental domain lies not so much with the definition of the set per se, but rather with how to treat integrals over the fundamental domain, when integrating functions with poles and zeros on the boundary of the domain.