---
tag: KNJA2NEZ9F
kind: concept
name: Fundamental domain
msc: ["22F10", "28C10", "22E40"]
sources:
  - url: https://en.wikipedia.org/wiki/Fundamental_domain
same_as:
  wikidata: Q1474108
  wikipedia: Fundamental domain
---

## Summary

A fundamental domain for a [[group action|0WCY42EV9V]] on a space is a subset that contains one point of each [[orbit|0YQPZB5Q1G]].
Often this is only asked up to a negligible boundary.
Its translates by the group tile the space.
Fundamental domains turn questions about the quotient space into questions about a concrete piece of the original space.
For instance, they compute the volume of a quotient by a lattice.

## Explanations

### L1 human

The integers act on the real line by shifting, and the interval $[0, 1)$ contains exactly one point from each set $x + \Z$.
Its shifts $[n, n+1)$ cover the line without overlap, like tiles.
In the plane, the unit square plays the same role for shifts by $\Z^2$.
Studying the quotient, here a circle or a torus, is the same as studying one tile with its edges glued.
The modular group acting on the upper half-plane has a famous fundamental domain bounded by two vertical lines and an arc of the unit circle.