  We prove that if $X:M^n\to\mathbb{H}^n\times \mathbb{R}$, $n\geq 3$, is a an
orientable, complete immersion with finite strong total curvature, then $X$ is
proper and $M$ is diffeomorphic to a compact manifold $\bar M$ minus a finite
number of points $q_1, \dots q_k$. Adding some extra hypothesis, including
$H_r=0,$ where $H_r$ is a higher order mean curvature, we obtain more
information about the geometry of a neighbourhood of each puncture.
  The reader will also find in this paper a classification result for the
hypersurfaces of $\mathbb{H}^n\times \mathbb{R}$ which satisfy $H_r=0$ and are
invariant by hyperbolic translations and a maximum principle in a half space
for these hypersurfaces.
