  In this paper we study the moduli space of properly Alexandrov-embedded,
minimal annuli in $\mathbb{H}^2 \times \mathbb{R}$ with horizontal ends. We say
that the ends are horizontal when they are graphs of $\mathcal{C}^{2, \alpha}$
functions over $\partial_\infty \mathbb{H}^2$. Contrary to expectation, we show
that one can not fully prescribe the two boundary curves at infinity, but
rather, one can prescribe the bottom curve, but the top curve only up to a
translation and a tilt, along with the position of the neck and the vertical
flux of the annulus. We also prove general existence theorems for minimal
annuli with discrete groups of symmetries.
