Groups Lecture 22

in which we find out about permutation properties of Möbius maps and start looking at cross-ratios.

Today we looked 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 $\frac{1}{z}$ 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.

We then looked at the cross-ratio of four distinct points $z_1, z_2, z_3, z_4 \in \mathbb{C}_\infty$. We know there is a unique Möbius map $g$ sending $z_1$ to $\infty$, $z_2$ to $0$ and $z_3$ to $1$. Then the cross-ratio is $[z_1,z_2,z_3,z_4]=g(z_4)$, i.e. it is the image of $z_4$ under that unique Möbius map.  In particular, $[\infty,0,1,\lambda]=\lambda$. The formula we had for the unique Möbius map $g$ gives us a formula for the cross-ratio as well: $[z_1,z_2,z_3,z_4]=\frac{z_4-z_2}{z_4-z_1}\frac{z_3-z_1}{z_3-z_2}$, with the usual special cases for infinity. From this formula it is clear that the double transpositions fix the cross-ratio$[z_1,z_2,z_3,z_4]= [z_2,z_1,z_4,z_3]= [z_3,z_4,z_1,z_2]= [z_4,z_3,z_2,z_1]$. It is also very easy to show from our actual definition (rather than the formula) that Möbius maps preserve cross-ratio: for any Möbius map $f$, we have $[z_1,z_2,z_3,z_4]= [f(z_1),f(z_2),f(z_3),f(z_4)]$. This follows from the uniqueness of the fore-mentioned Möbius map: If $[z_1,z_2,z_3,z_4]=g(z_4)$, then $gf^{-1}$ sends $f(z_1)$ to $\infty$, $f(z_2)$ to $0$ and $f(z_3)$ to $1$, so $[f(z_1),f(z_2),f(z_3),f(z_4)]=gf^{-1}(f(z_4))$$=g(z_4)= [z_1,z_2,z_3,z_4]$. We will use this next time to see that four points in $\mathbb{C}_\infty$ lie on a circle or line if and only if their cross ratio is real.

Understanding today’s lecture

First a comment on the end of last time’s lecture which I forgot in the last blog: 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 $f(z)=\frac{1}{z}$ 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? 🙂

Understanding cross-ratios comes best if you look at them from both points of view: the definition via Möbius maps as well as the formula.

Preparing for Lecture 23

Next time we will do some fun extra things.

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.