Groups Lecture 22

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 an inversion ( $f(z)=\frac{1}{z}$).  We called this the “geometry of Möbius maps“.

At the end of the lecture 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.

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 inversion, 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 inversion separately first to see if that gives you some ideas.

Preparing for Lecture 23

We will do more on fixed points next time, and we will cleverly use the conjugacy classes of Möbius maps for this to save us an awful lot of work. You can look up about the Jordan Normal Form in $2\times 2$ matrices again, either if you’ve done it in V+M by now, or from when I mentioned it in the Matrix Groups chapter. We’ll be using those to prove that every Möbius map (other than the identity) has exactly one or two fixed points. We will then 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.

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 we may or may not get to in the last lecture… If we don’t get to it, I’ll try to expand on the blog.

3 thoughts on “Groups Lecture 22”

1. Julia Goedecke

Gareth Taylor pointed me towards this very pretty video which shows visually why looking at Möbius maps as transformations of the Riemann Sphere is a good thing to do.

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2. Indi Pritchard

Small typo in the second main paragraph: cz+b instead of cz+d 🙂

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• Julia Goedecke

Thank you! I’ve corrected it now.

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