How to draw fundamental domain for a group generated by 2 elements acting on upper hlf plane

Question

It is easy to draw a fundamental domain for a cyclic group as in Chapter 1 of the book: "Geometry and Spectra of Compact Riemann Surfaces" by Peter Buser. However, I am very confused when trying to find a fundamental domain for a group generated by 2 elements for instance: $z \mapsto \frac{4z+1}{3z+1}$ and $z \mapsto \frac{5z+1}{4z+1}$. I hope someone will give me a hint. Thanks in advance

Answer 1

This is actually a rather deep problem. For example, consider a different example given by \begin{align*} f(z) &= 100 z \\ g(z) &= \frac{100z+1}{1+100z} \end{align*} (Notice that I have not bothered to normalize these to have determinant 1, the way you did in your question). For this example, since the fixed points of $f$ are $0$ and $\infty$, and since the fixed points of $g$ are $-1$ and $+1$, and since $100$ is a giganto-enormous number, it's not too hard to verify that there is a fundamental domain bounded by three tiny semicircles surrounding the three respective points $-1$, $0$, $+1$ and one gigantic semicircle centered on the point $0$.

But here's the problem. This pair of elements $f,g$ generate a free group $\langle f,g \rangle$ of rank 2. There are infinitely automorphisms of a free group of rank 2, and hence infinitely many 2 element generating sets of this group. Every one of those generating pairs generates the same group with the same fundamental domain. For instance, the two elements $f^2 g f^2 g fg$ and $f^2 g fg$ also generate the free group $\langle f,g \rangle$. So whatever procedure you come up with, when applied to $f^2 g f^2 g fg$ and $f^2 g fg$, will have to produce the same fundamental domain as when applied to $f$ and $g$.

What's even worse, there are many, many 2-generator subgroups of $PSL_2 \mathbb{R}$ (the group of Möbius transformations of the upper half plane) which are not discrete and therefore do not possess a fundamental domain. Whatever procedure you come up with will have to be know how to tell the difference between the discrete and nondiscrete cases, otherwise your procedure may go on forever.

This topic has been investigated rather thoroughly. If you are interested, you might take a look at the paper of Gilman and Maskit entitled "An algorithm for 2-generator Fuchsian groups". This paper describes an algorithm for deciding whether 2 given elements of $PSL_2\mathbb{R}$ generate a discrete group. The way the algorithm works is to first use the given pair of elements to test for one of two possibilities: either one can verify an obvious indiscreteness criterion known as Jorgensen's inequality; or one can verify discreteness by constructing an obvious fundamental domain. If neither of these possibilities holds, what the algorithm does is to compute another "simpler" generator pair. The key point is that as this algorithm proceeds, constructing "simpler" and "simpler" generator pairs, it will eventually stop at a generator pair for which either Jorgensen's inequality is verified, or an obvious fundamental domain is constructed.