Hyperbolic translation length ($\ell (A)=2\mathrm{arcosh}(|\mathrm{tr}A|/2)$)

For a hyperbolic Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$, the infimum of $d(z,Az)$ is attained on its axis and equals $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. It is the length of the corresponding closed geodesic in a quotient.