# Groups Lecture 23

in which we find out about fixed points and permutation properties of Möbius maps.

After seeing last time that any Möbius map with three fixed points must be the identity, we explored conjugacy classes of Möbius maps today. Any Möbius map is conjugate to either $f(z)=\nu z$ for $\nu \neq 0$, or to $f(z)=z+1$. We proved this via matrices, using the fact that every invertible $2\times 2$ matrix is conjugate to one in Jordan Normal Form. Then we saw that conjugate maps  have the same number of fixed points, so that means we know that any Möbius map has exactly one (like $f(z)=z+1$) or two (like $f(z)=\nu z$) fixed points.

This lead us to look at permutation properties of Möbius maps. Any Möbius map is determined by the images of three distinct points. And in fact, the Möbius group $M$ acts sharply three-transitively on the Riemann Sphere $\mathbb{C}_\infty$. This means that for any two triples $z_1,z_2,z_3$ and $w_1,w_2,w_3$ of distinct points in $\mathbb{C}_\infty$, there exists a unique Möbius map $f$ such that $f(z_i)=w_i$.

Since three points also determine a line or circle, we saw that Möbius maps send circles/straight lines to circles/straight lines. A better way to say it is that Möbius maps send circles on the Riemann Sphere to circles on the Riemann Sphere. This is because straight lines in $\mathbb{C}$ are circles through $\infty$ (or the North Pole) on the Riemann Sphere. We proved this either by using the general formula for circles/straight lines $Az\overline{z}+\overline{B}z +B\overline{z} +C=0$ with $A,C\in \mathbb{R}$ and $|B|^2>AC,$ or by using the geometry of Möbius maps and checking for rotation/dilation, translation and inversion separately. As an example we showed that $f(z)=\frac{z-i}{z+i}$ sends the real line to the unit circle, and the upper half plane to the inside of the circle. This is because complementary components are mapped to complementary components.

Understanding today’s lecture

It is really quite amazing that we only get these two types of conjugacy classes for Möbius maps. To understand what is going on, you could find out what map the inversion $f(z)=\frac{1}{z}$ is conjugate to. We already saw why the $f(z)=z+b$ becomes just $f(z)=z+1$, but it is worth going over that again and appreciating how amazing it is. The fact that conjugates have the same number of fixed points is very very useful. You might have an initial shyness to the term “sharply three-transitive”, but when you look at what it means, it isn’t so bad, is it? 🙂

Preparing for Lecture 24

Next time will be the last lecture! We will explore cross-ratios of complex numbers. Again that has much to do with geometry. We will define the cross-ratio of four (distinct) complex numbers, prove that Möbius maps preserve them, and see the connection between cross-ratios being real and four points lying on a circle/straight line. I expect we’ll have a bit of time for something extra, like projective spaces. There will also be a little video.

Going a little deeper

We showed today that Möbius maps take circles/straight lines to circles/straight lines. I mentioned without proof that complementary components are mapped to complementary components. This is a lot easier to show when you have the concept of continuity and some related things. In fact, Möbius maps preserve even more: they are so-called conformal maps, they preserve angles (with direction). This will be important in (the area of, or the course) Complex Analysis (or the course Complex Methods). So that means if two lines or circles meet at say a right angle, then they get mapped to two lines/circles which also meet at a right angle, and the direction of the angle stays the same. This means you can do arguments something like this: given our example $f(z)=\frac{z-i}{z+i}$ , when we “walk” from the point $\infty$ to $0$ and then to $1$ (in that order), the upper half plane is on our left hand side. So then once we’ve mapped these points to the unit circle, when we “walk” along the unit circle from $f(\infty)=1$ to $f(0)=-1$ to $f(1)=-i$, on our left hand side we have the inside of the circle, so that is where the upper half plane is mapped to. For now you can only really use this as a “sanity check” for your calculations, or to give you an indication/intuition on what is happening, but once you’ve formally proved in second year that Möbius maps are conformal, you can actually use it properly rigorously.

1. I was looking for a while for some little animations of stereographic projection and some Möbius maps, because I knew I had seen them somewhere. Finally found them, from a summer school I did in 2000. I’m sorry, I know it is in German, but the actual animations are language-free! Not quite as good as the one in the comment from last lecture, but it has some more functions if you’re interested. Some, like $f(z)=z^2$ or the exponential function, we haven’t done, but you might be interested anyway.