Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 203 / 1 / c / Solution

The assertion is false. Here $\mathbb{H}\setminus S=\{z:\mathrm{Im}z>1\}$. If a normalized map $g_S$ existed, thenwould be a conformal automorphism of $\mathbb{H}$. Hencewith real coefficients and $ad-bc=1$. Hydrodynamic normalization forces $h(w)\to \infty$ linearly, so $c=0$ and $h(w)=\alpha w+\beta$ with $\alpha >0$ and $\beta \in \mathbb{R}$. Butand $g_S(z)-z\to 0$ would require $\alpha =1$ and $\beta =i$, contradicting $\beta \in \mathbb{R}$.