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### Problem 1

The map $\Phi(z) = i\dfrac{1+z}{1-z}$ is a biholomorphic mapping from the unit disk $\mathbb D$ onto the upper half-plane $\mathbb H$. $\Phi$ is called the Cayley transform.

### Problem 2

Let $A=\begin{pmatrix} a & b\\c & d \end{pmatrix}\in SL(2,\mathbb R)$ and $\phi_A$ the Möbius transformation $\phi_A(z) = \dfrac{az+b}{cz+d}$. Then $\phi_A\in\text{Aut}(\mathbb H)$.

### Problem 3

For $a\in\mathbb H$, give an explicit $A\in SL(2,\mathbb R)$ so that $\phi_A$ takes $a$ to $i$.

### Problem 4

If $\psi\in\text{Aut}(\mathbb H)$, then there exists $A\in SL(2,\mathbb R)$ so that $\psi = \phi_A$.
(Hint: Consider the matrix $A = \begin{pmatrix} \cos\theta & -\sin\theta\\\sin\theta & \cos\theta \end{pmatrix}$ and verify that $\Phi^{-1}\circ\phi_A\circ\Phi$ is a rotation in $\mathbb D$. Use the previous problem and the discussion in class, with mappings fixing $0$ in $\mathbb D$ replaced by mappings fixing $i$ in $\mathbb H$.)

### Problem 5

Discuss the "uniqueness" of the matrix $A$ in theprevious problem, and thus find a subgroup $N\subset SL(2,\mathbb R)$ such that
$\text{Aut}(\mathbb H)\cong SL(2,\mathbb R)/N$.