Groups Lecture 21

in which we meet the Möbius group as transformations of the Riemann Sphere.

After finishing the duality between cube and octahedron, we looked at the symmetries of a tetrahedron and found that the rotations give us A_4 and the group of all symmetries is S_4. Notice that the tetrahedron does not have the “reflection in mid-point” as a symmetry, because the vertices are opposite midpoints of faces rather than other vertices.

As our last chapter, we started looking at Möbius maps. A Möbius map is f\colon {\mathbb{C}_\infty \to \mathbb{C}_\infty} of the form f(z)=\frac{az+b}{cz+d} for a,b,c,d\in \mathbb{C} and ad-bc \neq 0. This condition is needed to make the map non-constant, and indeed injective on \mathbb{C}. To include f(-\frac{d}{c}), we added a point at infinity. So we think of \mathbb{C}_\infty as the Riemann Sphere via the Stereographic Projection: take the unit sphere, cut it in the equator with the complex plane. Then attach a rod at the North Pole.  If we move around the rod, keeping it fixed at the North Pole, we see it can meet any point z in the complex plane. Then it intersects one more point on the sphere, and this is where we map the number z. Then the North Pole corresponds to \infty. Doing it this way, we have the unit circle exactly at the equator of the sphere, and everything inside the unit disc is in the southern hemisphere, while everything outside the unit circle is in the northern hemisphere.

Equipped with this extra understanding, we saw that Möbius maps are bijections from the Riemann Sphere to itself, and form a group, the Möbius group M. We will finish the proof that it really is a group next time.

Understanding today’s lecture

To understand the symmetries of regular polyhedra, the best way is to get one and play around with it. If you explore it for yourself, with the lecture notes as guide and “hints”, you will learn it much better than if you just take my word for it.

For the Möbius maps, you might take a little time to get used to the point at infinity. We will see next time that, although we often have to write out special checks for that point, it is not really any different. If you are afraid that you will have to do a lot of fraction manipulation which will be tedious, don’t worry: we will get a short cut next time via matrices.

This area of the course is very well represented and explained in Beardon Algebra and Geometry, so do consider using that book as another source.

Preparing for Lecture 22

After finishing the proof that the Möbius maps really form a group, we will look at them using matrices. You might be able to guess how that will work already. Using a surjective homomorphism from \mathrm{GL}_2(\mathbb{C}), we will also find that the Möbius group is isomorphic to the projective general linear group (and also to the projective special linear group, which happens to be the same for \mathbb{C}). We will show that any Möbius map is the composition of dilation/rotations, translations and inversions. After that we will look at fixed points of Möbius maps, and their conjugacy classes.

Going a little deeper

I mentioned as an extra that the Riemann Sphere is also called the one-point compactification of \mathbb{C}. “One-point” is clear, as we are adding only one point, \infty. What does compact mean? You will learn it in Metric and Topological spaces. For subsets of \mathbb{R}^n, it means that the set is closed and bounded. Bounded means that the distances between points in the set can’t go to infinity. A bounded set is closed if “it has a sharp edge”. For example, the interval [0,1]\subset \mathbb{R} is closed, but (0,1) is not. The unit sphere, seens as a subset of \mathbb{R}^3, is compact = closed and bounded. The proper definition of compact is that every open cover has a finite subcover, so again it is something to do with finiteness.

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One thought on “Groups Lecture 21

  1. I had some questions about the Riemann Sphere after the lecture. Why do we add the point at infinity to make a sphere, rather than just adding it as an “unconnected” point? The answer is quite complicated, I suppose.
    One place to look for explanations would be Complex Analysis. We need the point at infinity to be “close” to large complex numbers.
    Then there are geometrical considerations. The Riemann Sphere is a complex manifold, it is a Riemann surface, and lots of other things.
    If you are interested in either of those two viewpoints, you can find books in the schedules under the IB course “Complex Analysis” and the Part II course “Riemann Surfaces”. In fact there is one by Beardon which might be worth trying if you like the Beardon Algebra and Geometry book.
    Certainly, \mathbb{C}_\infty is not a field!

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