 Hyperbolic element of PSL2(R)

A real Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$ with $|\mathrm{tr}A|>2$. It has two fixed ideal boundary points and translates along the geodesic joining them; its hyperbolic translation length is $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. Conjugating by a real Möbius transformation puts it in the form $z\mapsto e^{\ell }z$ with $\ell >0$; its axis is then the imaginary axis.