Kaehler groups, quasi-projective groups, and 3-manifold groups

We prove two results relating 3-manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3-manifold. If N has non-empty, toroidal boundary, and \pi_1(N) is a Kaehler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or toroidal boundary, and \pi_1(N) is a quasi-projective group, then all the prime components of N are graph manifolds.