Fundamental area

Given a topological space and a group of actions on it, the images of a single point under the action of a group of actions form orbits of actions. A fundamental area is a subset of space that contains exactly one point from each orbit. It gives a geometric realization of an abstract set of orbit representatives.

There are many ways to choose a fundamental area. It is usually required that the fundamental region be a connected subset with some restrictions on the boundaries, for example, that they are smooth or multifaceted. The images of the selected fundamental region under the action of the group form a mosaic in space. One of the main constructions of fundamental areas is based on Voronoi diagrams .

Content

Hints on a general definition

The lattice on the complex plane and its fundamental region (factor space - torus).

If an action of a group G on a topological space X is given by means of homeomorphisms , the fundamental domain for such actions is the set D of representatives of the orbits. It is usually required that this set be topologically simple and be specified in one of several specific ways. The usual condition is that D be an almost open set in the sense that D must be the symmetric difference of an open set in G with a set of zero measure for some (quasi) invariant measure on X. The fundamental domain always contains a U , an open set that moves by the action of G into disconnected copies and in almost the same way as D represents orbits. It is often required that D be a complete set of representatives of adjacent classes with some repetitions, but that the repeating part has zero measure. This is a common situation in ergodic theories . If the fundamental domain is used to calculate the integral on X / G , the set of zero measure does not play a role.

For example, if X is a Euclidean space R ^{n of} dimension n and G is a lattice Z ⁿ acting on it as a parallel translation , the quotient space X / G is an n- dimensional torus . You can take D [0,1) ⁿ as a fundamental domain, which differs from an open set (0,1) ⁿ by a set of zero measure, or a closed unit cube [0,1] ⁿ whose boundary consists of points whose orbits have more than one representative in D.

Examples

Examples in the three-dimensional Euclidean space R ³ .

for n- fold rotation: the orbit consists of either n points around the axis, or a single point on the axis; fundamental area - sector
for specular reflection relative to the plane: the orbit consists of either two points, one on different sides of the plane, or a single point on the plane; fundamental area - half-space bounded by this plane
for central symmetry: the orbit consists of two points on opposite sides of the center, with the exception of a single orbit of the center itself; fundamental area - any half-space bounded by a plane passing through the center
to rotate 180 ° about the axis: the orbit consists of either two points located on opposite sides of the line, or of a single point on the line itself; fundamental region - any half-space bounded by a plane passing through the axis of symmetry
for discrete parallel transport in one direction: orbits form a one-dimensional lattice in the direction of the transport vector; fundamental region - an infinite region between two parallel planes
for discrete parallel transport in two directions: orbits form a two-dimensional lattice in the directions of transport vectors; the fundamental region has a parallelogram section
for discrete parallel transport in three directions: orbits form a lattice; the fundamental region is an elementary cell , which is, for example, a parallelepiped , or a Wigner-Seitz cell , which is also called a Voronoi cell / diagram .

In the case when parallel transport is combined with other types of symmetries, the fundamental region will be part of the unit cell. For example, for fundamental region is 1, 2, 3, 4, 6, 8, or 12 times smaller than the primitive cell.

Fundamental region of a modular group

The diagram on the right shows the part of constructing the fundamental domain for the action of the modular group Γ on the upper half-plane H (here, by the upper half-plane we mean the part of the complex plane with a positive coefficient for i ).

Any triangular region is a free regular set H / Γ. The gray region (with the third point at infinity) is the canonical fundamental region.

This famous chart appears in all classic books on modular functions . (Perhaps it was well known to Gauss , who studied fundamental areas in the study of the reduction of quadratic forms.) Here, each triangular region (bounded by blue lines) is a actions of Γ on H. Boundaries (blue lines) are not parts of free regular sets. To construct the fundamental region H / Γ, one needs to decide how to assign points on the boundaries, and one must be careful not to include these points twice. So, the free regular set for this example is

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

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

The fundamental region is constructed by adding the left border, plus half the arc below, including the midpoint:

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,\,{\frac {-1}{2}}<{\mbox{Re}}(z)\leq 0\right\}.

{\ displaystyle 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, \, {\ frac {-1} {2}} <{\ mbox {Re}} (z ) \ leq 0 \ right \}.}

The choice of which points to include varies from author to author.

The main difficulty in determining the fundamental domain does not lie directly in the definition of the set, but rather in how to work with integrals over the fundamental domain, when the integrands have poles and zeros on the boundary of the domain.

Links

Weisstein, Eric W. Fundamental domain on the Wolfram MathWorld website.