Cohomology groups with compact support for flat line bundles on certain complex Lie groups

Let $X$ be a complex surface obtained as the quotient of the complex Euclidean space ${\mathbb{C}}^2$ by a discrete subgroup of rank $3$. We investigate the cohomology group $H_0^1(X,E)$ with compact support for a unitary flat line bundle $E$ over $X$. We show the vanishing of $H_0^1(X,E)$ for a certain class of such pairs $(X,E)$, which includes infinitely many examples such that $H^1(X,E)$ is non-Hausdorff and infinite dimensional.