A fundamental domain for a group action of by homeomorphisms on a topological space , under a common closed-domain convention, is a subset that is the closure of its interior and whose translates cover ,

with whenever . Here denotes the interior of , and is the image of under (Voight 2021, p. 631). Thus distinct translates of the same fundamental domain have disjoint interiors, but they may meet on their boundaries.

For example, a fundamental domain of the group of rotations by multiples of in is the closed upper half-plane . For rotations by multiples of , the first quadrant, including its boundary, is a fundamental domain.

The group acting on by translations has the fundamental domains and . Their interiors overlap. The nonoverlap condition concerns translates of a single chosen fundamental domain, not two different fundamental domains. A closed fundamental domain may contain multiple representatives of a group orbit on its boundary. Requiring exactly one representative of every group orbit uses a different boundary convention.

This differs from a group block, whose translates are either identical or disjoint as sets. Translates of a closed fundamental domain may instead overlap on their boundaries.

See also

Brillouin Zone, Fundamental Parallelepiped, G-Set, Group Action, Group Block, Group Orbit

Portions of this entry contributed by David Terr

Portions of this entry contributed by Richard Peterson

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References

Voight, J. Quaternion Algebras. Cham, Switzerland: Springer, 2021. https://doi.org/10.1007/978-3-030-56694-4.

Referenced on Wolfram|Alpha

Fundamental Domain

Cite this as:

Weisstein, Eric W., with contributions by David Terr and Richard Peterson. "Fundamental Domain." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FundamentalDomain.html

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