A note on the volume flux group of four-manifolds

We show that closed orientable smooth four-manifolds with non-trivial volume flux group and fundamental group of subexponential growth type are finitely covered by a manifold homeomorphic to S 3 × S 1 , S 2 × T 2 or a nil-manifold. We also show that if a compact complex surface has non-trivial volume flux group then it has zero minimal volume.