  We study finite orbits for non-elementary groups of automorphisms of compact
projective surfaces. In particular we prove that if the surface and the group
are defined over a number field k and the group contains parabolic elements,
then the set of finite orbits is not Zariski dense, except in certain very
rigid situations, known as Kummer examples. Related results are also
established when k=C. An application is given to the description of "canonical
vector heights" associated to such automorphism groups.
