Define the modular group $\Gamma$ acting on the upper half-plane.

Describe the set $S$ of points $z$ in the upper half-plane that have $\operatorname{Im}(T(z)) \leqslant \operatorname{Im}(z)$ for each $T \in \Gamma$. Hence find a fundamental set for $\Gamma$ acting on the upper half-plane.

Let $A$ and $J$ be the two Möbius transformations

$$A: z \mapsto z+1 \quad \text { and } \quad J: z \mapsto-1 / z$$

When is $\operatorname{Im}(J(z))>\operatorname{Im}(z) ?$

For any point $z$ in the upper half-plane, show that either $z \in S$ or else there is an integer $k$ with

$$\operatorname{Im}\left(J\left(A^{k}(z)\right)\right)>\operatorname{Im}(z)$$

Deduce that the modular group is generated by $A$ and $J$.