  We study the hyperbolicity properties of the action of a non-elementary
automorphism group on a compact complex surface, with an emphasis on K3 and
Enriques surfaces. A first result is that when such a group contains parabolic
elements, Zariski diffuse invariant measures automatically have non-zero
Lyapunov exponents. In combination with our previous work, this leads to simple
criteria for a uniform expansion property on the whole surface, for groups with
and without parabolic elements. This, in turn, has strong consequences on the
dynamics: description of orbit closures, equidistribution, ergodicity
properties, etc.
  Along the way, we provide a reference discussion on uniform expansion of
non-linear discrete group actions on compact (real) manifolds and the
construction of Margulis functions under optimal moment conditions.
