Can someone explain the concept of Fundamental Domain to me?

Question

Hi for a Geometries course we're dealing with fundamental and I don't quite understand the definition of a fundamental domain.

The definition in my book is that a fundamental domain: is a region of a space of the quotient set such that it contains a representative of each equivalence class of the quotient and that there at most 1 representatives in its interior

The book didn't really give an example to illustrate this, can someone help out? What would be a simple example? We've covered Möbius geometries and its sub-geometries like hypberbolic and elliptic so an example in that context might also be useful.

Answer 1

When a group $G$ acts on a space $X$, it partitions the space into orbits:

$Orb(x):=\{g\cdot x\mid g\in G\}$

It is possible to choose a subset $S\subseteq X$ such that $\cup_{x\in S}Orb(x)=X$ in many different ways.

Firstly, if $a,b\in Orb(x)$ then you don't want to pick both $a$ and $b$ to be in $S$, because $Orb(a)=Orb(b)=Orb(x)$. So ideally we only want one thing per orbit (=partition).

Secondly, it's nice to require some additional topological requirements on the set of representatives you pick. For example, you might have $X=\Bbb R^2$ and maybe you can pick the representatives so that they're regularly spaced.

Once you have a representative of an orbit, you can get any other element from the orbit with a group action. This makes the set of representatives sort of like a basis of a vector space (loose analogy only).

The wiki page on fundamental domains already has several concrete examples you will probably be interested in.

Answer 2

The simplest example is probably the following:

The circle can be defined as the quotient of the real line by the equivalence relation defined by $x \cong x+1$. Then $[0,1)$ is a fundamental domain.

Going up a dimension, the torus is $\mathbb{R}^2$ mod the equivalence relation $(x,y) \cong (x+1,y) \cong(x,y+1)$. For this the unit square is a fundamental domain.

More fun is to look at various higher genus curves as quotients of the poincare disk by Fuchsian groups, and look at fundamental domains for those.