$\begingroup$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.
$\endgroup$2 Answers
$\begingroup$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.
$\endgroup$$\begingroup$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.
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