Consider a complex projective surface $X$, with a non-abelian free group $G$ acting faithfully and regularly on $X$. It may happen that $G$ has infinitely many periodic orbits: this is the case when $X$ is an abelian surface and all torsion points are $G$-periodic. In this talk, I will describe recent results obtained with Romain Dujardin aiming at a complete classification of all such examples. The main players will be canonical heights, arithmetic equidistribution, and rigidity results in ergodic theory.