  Examples of complete minimal surfaces properly embedded in H^2 x R have been
extensively studied and the literature contains a plethora of nontrivial ones.
In this paper we construct a large class of examples of complete minimal
surfaces embedded in H^2 x R, not necessarily proper, which are invariant by a
vertical translation or by a hyperbolic or parabolic screw motion. In
particular, we construct a large family of non-proper complete minimal disks
embedded in H^2 x R invariant by a vertical translation and a hyperbolic screw
motion and whose importance is twofold. They have finite total curvature in the
quotient of H^2 x R by the isometry, thus highlighting a different behaviour
from minimal surfaces embedded in R^3 satisfying the same properties. And they
show that the Calabi-Yau conjectures do not hold for embedded minimal surfaces
in H^2 x R.
