Fundamental domain
of a discrete group of transformations of a topological space
A subset containing elements from all the orbits (cf. Orbit) of
, 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
-algebra (for example, the Borel
-algebra) and containing exactly one representative from each orbit. However, if
is a topological manifold, then a fundamental domain is usually a subset
that is the closure of an open subset and is such that the subsets
,
, have pairwise no common interior points and form a locally finite covering of
. For example, as a fundamental domain of the group of parallel translations of the plane
by integer vectors one can take the square
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The choice of a fundamental domain is, as a rule, non-unique.
Comments
The chambers of the Weyl group are examples of fundamental domains of
in its reflection representation.
Fundamental domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_domain&oldid=47023