The group of hyperbolic motions is defined as the subgroup of S($\mathbb{H}^{2}$) (is the symmetric group of $\mathbb{H}^{2}$) which is generated by PGL($\mathbb{R})^{+}$ (matrices with positive determinate) and z$\mapsto \frac{1}{\overline{z}}$.
In the lecture, we also had that any hyperbolic motion is an automorphism of the hyperbolic plane as incidence geometry (for example, a straight line maps back into itself).
Can someone please give me an example?