A Möbius transformation is a map of the form
where is the extended complex plane and the ‘point at infinity’ is defined so that
- if then and ;
- if then .
The following video gives a very illuminating illustration. The sphere in the video is called the Riemann sphere, which in a sense ‘wraps up’ the extended complex plane into a sphere. Each point on the sphere corresponds to a unique point on the plane (i.e. there is a bijection between points on the extended plane and points on the sphere), with the ‘light source’ being the point at infinity. This bijective correspondence is the main reason for including the point at infinity.
According to the video any Möbius transformation can be generated by the four basic ones: translations, dilations, rotations and inversions:
- Translation: ,
- Dilation: ,
- Rotation: ,
Exercise. Show that any Möbius transformation is a composition of these four operations.
The Möbius transformations in fact form a group under composition which acts on . Moreover, we have a surjective homomorphism
Möbius transformations exhibit very interesting properties, some of which are:
Proposition 1. Given distinct , there is a unique Möbius map such that
Proof. It is not difficult to work out that the unique is given by
Proposition 2. The action of on is sharply triply transitive: if are distinct and are distinct, then there eixsts a unique such that for .
Proof. By proposition 1, there is a unique such that
and a unique such that
Then is the unique map satisfying the required property.
The cross-ratio of four distinct points is defined to be the unique such that if is the unique map satisfying
then , i.e.
One nice thing about cross-ratios is that they are preserved by Möbius transformations.
Proposition 3. If , then .
Proof. Let such that
Then . Likewise, if satisfies
then by proposition 1, so .
From we observe that some permutations of leave the value of the cross-ratio unaltered, e.g. is one such. What about the others?
Let act on the indices of the cross-ratio . The permutations that fix form the stabiliser subgroup of of this action. Using transitivity and invariance (propositions 2 and 3), the orbit of is just the different assignments of the values to ; i.e. the distinct cross-ratios that we get by permuting the indices are just
, , ,
, , .
Let such that
Then writing shows that the values of the above cross-ratios are (not necessarily in this order—too lazy to work out the precise order)
and they (as functions of ) form the subgroup of that fixes the set . This group is isomorphic to .
So there are in fact four permutations such that
and they form a subgroup of that is isomorphic to the Klein four-group