  We prove some existence and non-existence results for complete area
minimizing surfaces in the homogeneous space $\mathbb{E}(-1,\tau)$. As one of
our main results, we present sufficient conditions for a curve $\Gamma$ in
$\partial_{\infty} \mathbb{E}(-1,\tau)$ to admit a solution to the asymptotic
Plateau problem, in the sense that there exists a complete area minimizing
surface in $\mathbb{E}(-1,\tau)$ having $\Gamma$ as its asymptotic boundary.
