Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 4 / 11E / a / Solution

The disc model isand the upper half-plane model isThe Cayley mapmaps $D$ bijectively to $\mathfrak{h}$. Since $C^{\prime }(z)=2i/(1-z{)}^2$ and $\mathrm{Im}C(z)=(1-|z{|}^2)/|1-z{|}^2$, direct substitution gives $C^{\ast }{g}_{\mathfrak{h}}=g_D$.

Representing $C$ by $K=\left(\begin{array}{cc}i& i\\ -1& 1\end{array}\right)$, the disc isometry corresponding to $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is represented, up to a nonzero scalar, by