Math 2.0

Topological groups, Lie groups Definition

Fundamental domain

Builds on Group actionMeasure spaceMeasure-preserving transformationCountable setOrbit

Definition

Let (X,𝒜,μ)(X, \mathcal{A}, \mu) be a measure space and let GG be a countable group that acts on XX by measure-preserving measurable maps x↦gxx \mapsto gx. A measurable set F⊆XF \subseteq X is a fundamental domain for this action if:

  1. the translates cover almost all of XX: μ(X∖⋃g∈GgF)=0\mu\big(X \setminus \bigcup_{g \in G} gF\big) = 0;
  2. distinct translates are almost disjoint: μ(gF∩hF)=0\mu(gF \cap hF) = 0 for all g,h∈Gg, h \in G with g≠hg \ne h.

Remarks

Example: G=ℤG = \Z acting on X=ℝX = \R with Lebesgue measure by n⋅x=x+nn \cdot x = x + n, and F=[0,1)F = [0, 1). The translates F+n=[n,n+1)F + n = [n, n+1) cover ℝ\R and are pairwise disjoint, so both clauses hold.

Non-example: for the same action, F=[0,2)F = [0, 2) covers ℝ\R but fails clause 2, since F∩(F+1)=[1,2)F \cap (F + 1) = [1, 2) has measure 11.

Without a measure, a strict fundamental domain is a set that meets each orbit in exactly one point. In geometry one often asks instead for a closed region DD whose translates cover XX and whose interiors are pairwise disjoint. Any two measurable fundamental domains of the same action have the same measure.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Countable set, concept, no evidence levelCountable setconceptGroup action, concept, no evidence levelGroup actionconceptMeasure space, concept, no evidence levelMeasure spaceconceptMeasure-preserving transformation, concept, no evidence levelMeasure-preserving tr…conceptOrbit, concept, no evidence levelOrbitconceptFundamental domain, definition, no evidence levelFundamental domaindefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Countable set concept
  2. Group action concept
  3. Measure space concept
  4. Measure-preserving transformation concept
  5. Orbit concept
  6. Fundamental domain definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_domain
About this page
TagWEGEXJBX26
URIm2://WEGEXJBX26
Kinddefinition
WritingW0 stub
MSC22F10, 28C10, 22E40
DefinesFundamental domain
UsesGroup action, Measure space, Measure-preserving transformation, Countable set, Orbit
Changed2 November 2026

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