Abstract
For each
$k\geq 3$
, we construct a
$1$
-parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space
$\mathbb {H}^2\times \mathbb {R}$
with genus
$1$
and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus
$1$
and
$2k$
ends in the quotient of
$\mathbb {H}^2\times \mathbb {R}$
by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature
$-4k\pi $
. Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus
$1$
in quotients of
$\mathbb {H}^2\times \mathbb {R}$
by the action of a hyperbolic or parabolic translation.