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Define $$\phi_a(z) = \frac{z-a}{1-\overline{a}z}, \qquad \rho_\alpha(z) = e^{i\alpha}z,$$ with $|a|<1$ and $\alpha \in \mathbb{R}$, so that $\phi_a \circ \rho_\alpha$ is a holomorphic automorphism of the unit disc.

There are then three questions:

(I) Find $c$ and $\gamma$ such that $(\phi_a \circ \rho_\alpha)\circ(\phi_b \circ \rho_\beta) = \phi_c \circ \rho_\gamma$.

(II) Compute $(\phi_a \circ \rho_\alpha - \rho_\alpha \circ \phi_a)(z) $.

(III) Using knowledge of the automorphisms of the unit disc, describe the automorphisms of the upper half plane $\mathbf{H}$ as fractional linear transformations.

I mistakenly thought that part (I) would come down to just some simple algebraic manipulation, but I've gone through about a dozen pages of attempts and I just can't seem to get it in the form required to extract a suitable $c$ and $\gamma$. I'm really hoping that there's some neat trick that will make the answer just fall out.

For part (II), I've gone through it and obtained a fairly horrendous answer, but I was wondering whether there is actually a tidy answer, namely one tidier than: $$(\phi_a \circ \rho_\alpha - \rho_\alpha \circ \phi_a)(z)=\frac{\overline{a} e^{i\alpha} z \left( e^{i 2 \alpha} z - z - a e^{i 2 \alpha}\right) + a \overline{a} z + a e^{i \alpha} - a}{1 - \overline{a}z - \overline{a}e^{i \alpha}z + (\overline{a})^2 e^{i \alpha} z^2}.$$

Finally, for (III), I looked it up and the automorphisms of the upper half plane are $$z \mapsto \frac{az+b}{cz+d}, \quad ad-bc=1.$$ However, my interpretation of the question is to map the upper half plane to the unit disc, apply an automorphism and then map it back, i.e. for any $f \in \text{Aut}(\mathbf{H})$, $f = (\xi^{-1} \circ g \circ \xi) (z)$. So what I did was define $\xi(z) = \frac{z-i}{z+1}$, $g(z) = e^{i \alpha} \frac{z-a}{1-\overline{a}z}$, and then of course $\xi^{-1}(z) = \frac{z+1}{i z - i}$. Unfortunately for me, though, when I perform this composition I can't get it in a form commensurate with what I already know to be the answer.

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Both the automorphisms of the unit disc and the automorphisms of the upper half plane are groups (under composition) that are isomorphic to groups of matrices (under matrix multiplication). So, when computing long compositions of transformations, it is the same to simply multiply the corresponding matrices. I think this may help with part (I). –  John Adamski May 7 '12 at 12:59
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Concerning (I) and (II), I second the advice given by John Adamski: it is easier to multiply matrices of the form $\begin{pmatrix} a & b \\c & d \end{pmatrix}$ then to compose maps of the form $\dfrac{az+b}{cz+d}$. The result is the same, see here.

Concerning (II), I am surprised that someone wanted to know $\phi_a \circ \rho_\alpha - \rho_\alpha \circ \phi_a$. Subtracting two automorphisms of the unit disk does not look like a natural operation to me. Maybe they meant the multiplicative commutator, $\phi_a \circ \rho_\alpha \circ (\rho_\alpha \circ \phi_a)^{-1}$?

So what I did was define $\xi(z)=\dfrac{z−i}{z+1}$

A typo here: $\xi(z)=\dfrac{z−i}{z+i}$ would be correct. Not sure if this is where the problem was. Anyway, as in (III) it is better to work with matrices and compute $$\begin{pmatrix} 1 & -i \\1 & i \end{pmatrix}^{-1} \begin{pmatrix} a & b \\c & d \end{pmatrix}\begin{pmatrix} 1 & -i \\1 & i \end{pmatrix} $$

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