Fundamental Domain

Let G be a group and S be a topological G-set. Then a closed subset F of S is called a fundamental domain of G in S if S is the union of conjugates of F, i.e.,

and the intersection of any two conjugates has no interior.

For example, a fundamental domain of the group of rotations by multiples of 180 degrees in R^2 is the upper half-plane {(x,y)|y>=0} and a fundamental domain of rotations by multiples of 90 degrees is the first quadrant {(x,y)|x,y>=0}.

The concept of a fundamental domain is a generalization of a minimal group block, since while the intersection of fundamental domains has empty interior, the intersection of minimal blocks is the empty set.

See also

Portions of this entry contributed by David Terr

Portions of this entry contributed by Richard Peterson

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Cite this as:

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

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