Groups Lecture 24

in which we learn about cross-ratio and the complex projective line.

We started with a little video and some animations demonstrating certain Möbius maps and the Stereographic Projection.

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]$. That immediately shows us that four points in $\mathbb{C}_\infty$ lie on a circle or line if and only if their cross ratio is real.

As a little extra, we looked at the complex projective line. This is the set of one-dimensional subspaces of $\mathbb{C}^2$, which we can represent by their “slopes”: $\langle\left(\begin{smallmatrix}z_1 \\ z_2\end{smallmatrix}\right)\rangle$ corresponds to $\frac{z_1}{z_2}$, which is a complex number when $z_2\neq 0$, or $\infty$ when $z_2=0$.

Understanding today’s lecture

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.

After a whole groups lecture course (and of course three other courses as well), you may be able to understand roughly half of the maths puns in this:

Finite Simple Group (of Order Two)

Listen to it again after some Analysis, and then after third year, and then… (there is at least one Category Theory joke in it).

Going a little deeper

There is a lot to say about projective spaces, but this blog is too small to contain them 🙂

If you want to test your understanding and memory of our groups lecture course, try this:

Question 7 on Rachel Camina’s third example sheet from last year. We have actually done all this in lectures, but not in the same place, so you’ll have to “pull the threads together”.

And if you have solved all my example sheet questions, here is a harder problem sheet by Gareth Taylor.