  We construct the first examples of complete, properly embedded minimal
surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and
positive genus. These are constructed by gluing copies of horizontal catenoids
or other nondegenerate summands. We also establish that every horizontal
catenoid is nondegenerate.
  Finally, using the same techniques, we are able to produce properly embedded
minimal surfaces with infinitely many ends. Each annular end has finite total
curvature and is asymptotic to a vertical totally geodesic plane.
