Practice problem

Complex Analysis and Riemann Surfaces

Recall the identification $z \mapsto \frac{z-i}{z+i}$ of the upper half plane $\mathbb{H}$ with the unit disk $\mathbb{D}$ . (Note: $z \mapsto e^{iz}$ is periodic, so it does not work.)

Describe the stabilizer of $i\in \mathbb{H}$ within $\operatorname{Aut}(\mathbb{H}) = \operatorname{PSL}_2(\mathbb{R})$ . Denote an element of $\operatorname{PSL}_2(\mathbb{R})$ by $[z:w]\mapsto [az+bw:cz+dw]$ .