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Difference between revisions of "Fundamental domain"

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''of a discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f0421901.png" /> of transformations of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f0421902.png" />''
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A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f0421903.png" /> containing elements from all the orbits (cf. [[Orbit|Orbit]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f0421904.png" />, with exactly one element from orbits in general position. There are various versions of the exact definition of a fundamental domain. Sometimes a fundamental domain is any subset belonging to a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f0421905.png" />-algebra (for example, the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f0421906.png" />-algebra) and containing exactly one representative from each orbit. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f0421907.png" /> is a topological manifold, then a fundamental domain is usually a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f0421908.png" /> that is the closure of an open subset and is such that the subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f0421909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f04219010.png" />, have pairwise no common interior points and form a locally finite covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f04219011.png" />. For example, as a fundamental domain of the group of parallel translations of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f04219012.png" /> by integer vectors one can take the square
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f04219013.png" /></td> </tr></table>
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''of a discrete group  $  \Gamma $
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of transformations of a topological space  $  X $''
  
The choice of a fundamental domain is, as a rule, non-unique.
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A subset  $  D \subset  X $
 +
containing elements from all the orbits (cf. [[Orbit|Orbit]]) of  $  \Gamma $,
 +
with exactly one element from orbits in general position. There are various versions of the exact definition of a fundamental domain. Sometimes a fundamental domain is any subset belonging to a given  $  \sigma $-
 +
algebra (for example, the Borel  $  \sigma $-
 +
algebra) and containing exactly one representative from each orbit. However, if  $  X $
 +
is a topological manifold, then a fundamental domain is usually a subset  $  D \subset  X $
 +
that is the closure of an open subset and is such that the subsets  $  \gamma D $,
 +
$  \gamma \in \Gamma $,  
 +
have pairwise no common interior points and form a locally finite covering of  $  X $.  
 +
For example, as a fundamental domain of the group of parallel translations of the plane  $  \mathbf R  ^ {2} $
 +
by integer vectors one can take the square
  
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$$
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\{ {( x, y) \in \mathbf R  ^ {2} } : {
 +
0 \leq  x \leq  1,\
 +
0 \leq  y \leq  1 } \}
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.
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$$
  
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The choice of a fundamental domain is, as a rule, non-unique.
  
 
====Comments====
 
====Comments====
The chambers of the [[Weyl group|Weyl group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f04219014.png" /> are examples of fundamental domains of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042190/f04219015.png" /> in its reflection representation.
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The chambers of the [[Weyl group|Weyl group]] $  W $
 +
are examples of fundamental domains of $  W $
 +
in its reflection representation.

Latest revision as of 19:40, 5 June 2020


of a discrete group $ \Gamma $ of transformations of a topological space $ X $

A subset $ D \subset X $ containing elements from all the orbits (cf. Orbit) of $ \Gamma $, with exactly one element from orbits in general position. There are various versions of the exact definition of a fundamental domain. Sometimes a fundamental domain is any subset belonging to a given $ \sigma $- algebra (for example, the Borel $ \sigma $- algebra) and containing exactly one representative from each orbit. However, if $ X $ is a topological manifold, then a fundamental domain is usually a subset $ D \subset X $ that is the closure of an open subset and is such that the subsets $ \gamma D $, $ \gamma \in \Gamma $, have pairwise no common interior points and form a locally finite covering of $ X $. For example, as a fundamental domain of the group of parallel translations of the plane $ \mathbf R ^ {2} $ by integer vectors one can take the square

$$ \{ {( x, y) \in \mathbf R ^ {2} } : { 0 \leq x \leq 1,\ 0 \leq y \leq 1 } \} . $$

The choice of a fundamental domain is, as a rule, non-unique.

Comments

The chambers of the Weyl group $ W $ are examples of fundamental domains of $ W $ in its reflection representation.

How to Cite This Entry:
Fundamental domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_domain&oldid=13590
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article