The upper half-plane automorphism group is $\PSL_2(\RR)$ acting by real Möbius transformations.Suggest a slogan
Let \(\HH\coloneqq\{z\in\CC : \operatorname{Im}z>0\}\) be the upper half-plane. The group \(\Aut_\CC(\HH)\) of biholomorphisms \(\HH\to\HH\) is \[ \Aut_\CC(\HH)=\Bigl\{z\mapsto\frac{az+b}{cz+d} : a,b,c,d\in\RR,\ ad-bc=1\Bigr\}, \] and sending \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\SL_2(\RR)\) to this map induces a group isomorphism \(\PSL_2(\RR)\cong\Aut_\CC(\HH)\).
See [@SS03].