Classification of Möbius transformations over the Poincaré disk [closed]

Question

I'm studying the space of relations with given size in the group of isometries of the Poincaré disk. The unitary disk in the complex plane with a suitable metric is the Poincaré disk, and on this model the (orientation-preserving) isometries are given by the Möbius transformations on the disk:

$$f(z)=e^{i\theta}\frac{z+a}{\bar az + 1}, |a|<1$$

Equivalently, we can write these transformations as

$$f(z) = \frac{cz+a}{\bar az +\bar c}, |a|<|c|$$

I'm trying to find a classification of these isometries in elliptic, parabolic and hyperbolic, which the respectively parameters.

For instance, is easy to see that $f(z)=\frac{z+a}{\bar az + 1}$ is a hyperbolic transformation since it fixes $\pm\frac{a}{|a|}$ and the parameter can be calculated knowing that $-a\mapsto0$.

I had some tries but it still far from complete. Any reference or tips?

Answer 1

One way to proceed is to use the following biholomorphism $$\mathcal{H}\rightarrow D,z\mapsto \frac{z-i}{z+i}$$, where $\mathcal{H}$ is the Poincaré upper half plane endowed with the hyperbolic metric $\frac{dx^2+dy^2}{y^2}$. The group of biholomorphisms of $\mathcal{H}$ is $\mathrm{SL}_{2}(Z)/<-\mathrm{I}_2>$ where a matrix $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ acts on $\mathcal{H}$ by fractional linear transformation $$\tau\mapsto \gamma.\tau=\frac{a\tau+b}{c\tau+d}.$$ The classification you ask is given directly by the trace of the matrix by a straightforward computation (solving the equation $\gamma.tau=\tau$):

1. If $|Trace(\gamma)|>2$ then the element is hyperbolic.
2. If $|Trace(\gamma)|=2$ then is parabolic.
3. Otherwise it is elliptic.

This is standard material and can be found for example in the first chapter of the book "The spectrum of hyperbolic surfaces" by Nicolas Bergeron.