Groups Lecture 21

in which we view Möbius maps via matrices, and compose geometrically meaningful maps.

We finished the proof that the Möbius maps form a group, and saw that it is a non-abelian group. The easiest way to deal with the special cases that come up around \infty is to use the conventions “\frac{1}{\infty}=0“, “\frac{1}{0}=\infty” and “\frac{a\infty}{c\infty}=\frac{a}{c}“. Looking at the Riemann Sphere, it is clear that infinity is not really a special point at all; it just happens to be a bit different when we come from the \mathbb{C} perspective rather than the Riemann Sphere perspective.

To increase our understanding of the Möbius group (and to save us lots of tedious fraction manipulations), we looked at Möbius maps via matrices: we proved that there is a surjective group homomorphism from the invertible 2\times 2 matrices to the Möbius maps: \theta\colon {\mathrm{GL}_2(\mathbb{C})\to M} sends \left(\begin{smallmatrix}a & b\\ c& d\end{smallmatrix}\right) to \frac{az+b}{cz+d}. It is clear that this is not injective: the kernel consists of all scalar matrices \lambda I, which is also the centre of \mathrm{GL}_2(\mathbb{C}).

We then proved that every Möbius map is a composite of maps which can be described as dilation/rotations (f(z)=az for a\neq 0), translations (f(z)=z+b) and a combined inversion plus reflection (f(z)=\frac{1}{z}).  We called this the “geometry of Möbius maps“. You can see a nice animation of these types in this You-tube video. Dilation is like moving the Riemann sphere up and down, rotation is rotation the Riemann sphere around the z-axis, translation is moving the Riemann sphere to a different point on the complex plane, and the inversion plus reflection \frac{1}{z} is rotating the Riemann sphere by 180^\circ around the x-axis.

After this we started on fixed points, and saw first that any Möbius map which has at least three fixed points is the identity. This is because non-trivial quadratics have one or two roots. To understand more about the fixed points of a Möbius map, we explored conjugacy classes of Möbius maps: any Möbius map is conjugate either to f(z)=\nu z for some \nu \neq 0, or to f(z)=z+1. We proved this using the known conjugacy classes (i.e. Jordan Normal Forms) of complex 2\times 2 matrices. Notice that the group hom from \mathrm{GL}_2(\mathbb{C}) to M gives that if two matrices are conjugate, the corresponding Möbius maps are also conjugate, but not the other way round: the matrices \left(\begin{smallmatrix}\lambda & 1\\ 0&\lambda\end{smallmatrix}\right) and  \left(\begin{smallmatrix}1 & \frac{1}{\lambda}\\ 0&1\end{smallmatrix}\right) are not conjugate as matrices, but do give the same Möbius map. 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.

Understanding today’s lecture

Working with Möbius maps, especially composing them and finding their inverses, is much easier when you use the matrix techniques. A lot of properties that we will be wanting to show can be shown more easily for the special cases of dilation/rotation, translation and \frac{1}{z}, and if they are properties that work well with composition, that will be enough. So if you don’t know how to start on something, perhaps try with dilation, translation and \frac{1}{z} separately first to see if that gives you some ideas.

Preparing for Lecture 23

Next time we will look at permutation properties of Möbius maps: the images of any three (distinct) points determine a Möbius map uniquely. Adding an existence proof to that will show us that Möbius maps act sharply three-transitively on the Riemann Sphere. (Now there is a wonderful expression 🙂 .) We shall also look at what Möbius maps do to circles on the Riemann Sphere. Then we might start on cross-ratios, which also has to do with circles and mappings of Möbius maps.

Going a little deeper

I mentioned more or less in passing that the Möbius group is isomorphic to the projective general linear group and the projective special linear group over \mathbb{C}. This already tells you that those two must be the same (well, isomorphic). But they are not over other fields, for example over \mathbb{R}. You can define them the same way: quotient out the non-zero scalar matrices. But if you do that for \mathrm{GL}_n(\mathbb{R}) and \mathrm{SL}_n(\mathbb{R}), you will get different groups. It all has to do with projective spaces, which you could look up if you like.

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