Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 2 / a / Solution

For fixed $\tau =x+iy$, the set $\{|c\tau +d|:(c,d)\in {\mathbb{Z}}^2\setminus \{0\}\}$ has a positive minimum. Choose a primitive pair $(c,d)$ attaining it and complete it to a matrix $\gamma \in S{L}_2(\mathbb{Z})$. Sincethis point has maximal imaginary part in the orbit. Translate by a power of $T:\tau \mapsto \tau +1$ to arrange $|\mathrm{Re}\tau |\le 1/2$. If now $|\tau |<1$, applying $S:\tau \mapsto -1/\tau$ strictly increases the imaginary part, a contradiction. Thus $|\tau |\ge 1$, proving that every orbit meets the standard fundamental domain of the modular group $\mathcal{F}$.