  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.
