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en>Anirudh Emani: Reverted edits by 82.132.234.83 (talk) (HG)

2014-11-20T18:53:42Z

Reverted edits by 82.132.234.83 (talk) (HG)

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	~~Given a [[topological space]]~~ and ~~a [[group (mathematics)\|group]] [[group action\|acting]] on~~ it~~, the images of a single point under the group action form an [[group action#Orbits_and_stabilizers\|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.~~		<br><br>Vicente precisely what you can call him and he believes it sounds quite suitable. My friends say it's not friendly to me but what Good doing is bird keeping but Do not think have the time lately. Rhode Island is the place where he loves most. Procuring is a few things i do as well as it something I actually enjoy. Go to my website to know the truth more: http://devolro.com/expeditions<br><br>Feel free to visit my blog ... [http://devolro.com/expeditions organizations]

	~~There are many ways~~ to ~~choose a fundamental domain. Typically, a fundamental domain~~ is required to be a [[connected space\|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 [[tessellation\|tile]] the space. One general construction of fundamental domains uses [[Voronoi cell]]s.

	~~== Hints at general definition ==~~
	~~[[Image:Lattice torsion points.svg\|right\|thumb\|300px\| A lattice in~~ the ~~complex plane and its fundamental domain, with quotient a torus~~. ]]
	Given an [[ group action\|action]] of a [[group (mathematics)\|group]] ''G'' on a [[topological space]] ''X'' by [[homeomorphism]]s, 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 (mathematics)\|measure]] on ''X''. A fundamental domain always contains a [[free regular set]] ''U'', an [[open set]] moved around by ''G'' into [[Disjoint sets\|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'''<sup>''n''</sup> of dimension ''n'', and ''G'' is~~ the ~~[[lattice (group theory)\|lattice]] '''Z'''<sup>''n''<~~/~~sup> acting on it by translations, the quotient ''X''~~/~~''G'' is the ''n''-dimensional [[torus]]~~. ~~A fundamental domain ''D'' here can be taken to be <nowiki>[0,1)<~~/~~nowiki>~~<~~sup~~>~~''n''~~<~~/sup~~>, which differs from the open set (0,1)<sup>''n''</sup> by a set of measure zero, or the [[closed set\|closed]] unit cube <nowiki>[0,1]</nowiki><sup>''n''</sup>, whose [[boundary (topology)\|boundary]] consists of the points whose orbit has more than one representative in ''D''.

	~~== Examples ==~~

	~~Examples in the three-dimensional Euclidean space '''R'''<sup>3</sup>.~~
	*for ''n''-fold rotation: an orbit is either a set of ''n'' points around the axis, or a single point on the axis; the fundamental domain is a sector
	*for reflection in a plane: an orbit is either a set of 2 points, one on each side of the plane, or a single point in the plane; the fundamental domain is a half-space bounded by that plane
	*for inversion in a point: an orbit is a set of 2 points, one on each side of the center, except for one orbit, consisting of the center only; the fundamental domain is a half-space bounded by any plane through the center
	*for 180° rotation about a line: an orbit is either a set of 2 points opposite to ~~each other with respect to the axis, or a single point on the axis; the fundamental domain is a half-space bounded by any plane through the line~~
	*for discrete [[translational symmetry]] in one direction: the orbits are translates of a 1D lattice in the direction of the translation vector; the fundamental domain is an infinite slab
	*for discrete translational symmetry in two directions: the orbits are translates of a 2D lattice in the plane through the translation vectors; the fundamental domain is an infinite bar with [[parallelogram]]matic cross section
	*for discrete translational symmetry in three directions: the orbits are translates of the lattice; the fundamental domain is a [[primitive cell]] which is e.g. a [[parallelepiped]], or a [[Wigner-Seitz cell]], also called [[Voronoi diagram\|Voronoi cell]]/diagram.

	~~In the case of translational symmetry combined with other symmetries, the fundamental domain is part of the primitive cell~~. ~~For example, for [[wallpaper group]]s the fundamental domain is a factor 1, 2, 3, 4, 6, 8, or 12 smaller than the primitive cell~~.

	~~== Fundamental domain for the modular group ==~~

	[[~~Image~~:ModularGroup-FundamentalDomain-01.png\|thumb\|400px\|Each triangular region is a free regular set of H/Γ; the grey one (with the third point of the triangle at infinity) is the canonical fundamental domain.]]
	~~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 [[modular function]]s. (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~~

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

	~~The fundamental domain is built by adding the boundary on the left plus half the arc on the bottom including the point in the middle:~~

	~~:<math>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\}~~.</~~math>~~

	~~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.

	~~== See also ==~~
	* [[Free regular set]]
	* [[Fundamental polygon]]
	* [[Brillouin zone]]
	* [[Fundamental pair of periods]]
	* [[Petersson inner product]]
	* [[Cusp neighborhood]]

	~~== External links ==~~
	* {{MathWorld \| urlname=FundamentalDomain \| title=Fundamental domain }}

	~~[[Category:Topological groups]]~~
	~~[[Category:Ergodic theory]]~~
	~~[[Category:Riemann surfaces]]~~
	~~[[Category:Group actions]~~]

82.150.99.2: Further clarify quoting convention

2013-10-29T19:04:18Z

Further clarify quoting convention

New page

Given a [[topological space]] and a [[group (mathematics)|group]] [[group action|acting]] on it, the images of a single point under the group action form an [[group action#Orbits_and_stabilizers|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 space|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 [[tessellation|tile]] the space. One general construction of fundamental domains uses [[Voronoi cell]]s.

== Hints at general definition ==
[[Image:Lattice torsion points.svg|right|thumb|300px| A lattice in the complex plane and its fundamental domain, with quotient a torus. ]]
Given an [[ group action|action]] of a [[group (mathematics)|group]] ''G'' on a [[topological space]] ''X'' by [[homeomorphism]]s, 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 (mathematics)|measure]] on ''X''. A fundamental domain always contains a [[free regular set]] ''U'', an [[open set]] moved around by ''G'' into [[Disjoint sets|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 (group theory)|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 <nowiki>[0,1)</nowiki>''n'', which differs from the open set (0,1)''n'' by a set of measure zero, or the [[closed set|closed]] unit cube <nowiki>[0,1]</nowiki>''n'', whose [[boundary (topology)|boundary]] consists of the points whose orbit has more than one representative in ''D''.

== Examples ==

Examples in the three-dimensional Euclidean space '''R'''3.
*for ''n''-fold rotation: an orbit is either a set of ''n'' points around the axis, or a single point on the axis; the fundamental domain is a sector
*for reflection in a plane: an orbit is either a set of 2 points, one on each side of the plane, or a single point in the plane; the fundamental domain is a half-space bounded by that plane
*for inversion in a point: an orbit is a set of 2 points, one on each side of the center, except for one orbit, consisting of the center only; the fundamental domain is a half-space bounded by any plane through the center
*for 180° rotation about a line: an orbit is either a set of 2 points opposite to each other with respect to the axis, or a single point on the axis; the fundamental domain is a half-space bounded by any plane through the line
*for discrete [[translational symmetry]] in one direction: the orbits are translates of a 1D lattice in the direction of the translation vector; the fundamental domain is an infinite slab
*for discrete translational symmetry in two directions: the orbits are translates of a 2D lattice in the plane through the translation vectors; the fundamental domain is an infinite bar with [[parallelogram]]matic cross section
*for discrete translational symmetry in three directions: the orbits are translates of the lattice; the fundamental domain is a [[primitive cell]] which is e.g. a [[parallelepiped]], or a [[Wigner-Seitz cell]], also called [[Voronoi diagram|Voronoi cell]]/diagram.

In the case of translational symmetry combined with other symmetries, the fundamental domain is part of the primitive cell. For example, for [[wallpaper group]]s the fundamental domain is a factor 1, 2, 3, 4, 6, 8, or 12 smaller than the primitive cell.

== Fundamental domain for the modular group ==

[[Image:ModularGroup-FundamentalDomain-01.png|thumb|400px|Each triangular region is a free regular set of H/Γ; the grey one (with the third point of the triangle at infinity) is the canonical fundamental domain.]]
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 [[modular function]]s. (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

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

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

:<math>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\}.</math>

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.

== See also ==
* [[Free regular set]]
* [[Fundamental polygon]]
* [[Brillouin zone]]
* [[Fundamental pair of periods]]
* [[Petersson inner product]]
* [[Cusp neighborhood]]

== External links ==
* {{MathWorld | urlname=FundamentalDomain | title=Fundamental domain }}

[[Category:Topological groups]]
[[Category:Ergodic theory]]
[[Category:Riemann surfaces]]
[[Category:Group actions]]