15E (source code)
= 15E
{scope}
{title2=Geometry}
The group
$$
PSL_2(\mathbb R)=SL_2(\mathbb R)/\{\pm I\}
$$
acts on the <Poincaré half-plane model> by
$$
g(z)=\frac{az+b}{cz+d},
\qquad ad-bc=1.
$$
A direct calculation gives
$$
\operatorname{Im}g(z)=\frac{\operatorname{Im}z}{|cz+d|^2},
\qquad
dg=\frac{dz}{(cz+d)^2}.
$$
Therefore
$$
\frac{|dg|^2}{(\operatorname{Im}g(z))^2}
=\frac{|dz|^2}{(\operatorname{Im}z)^2},
$$
so every element of <PSL2(R)> is an isometry of the hyperbolic metric.
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