  Let N be a closed, oriented 3-manifold. A folklore conjecture states that
$S^{1} \times N$ admits a symplectic structure if and only if $N$ admits a
fibration over the circle. We will prove this conjecture in the case when N is
irreducible and its fundamental group satisfies appropriate subgroup
separability conditions. This statement includes 3-manifolds with vanishing
Thurston norm, graph manifolds and 3-manifolds with surface subgroup
separability (a condition satisfied conjecturally by all hyperbolic
3-manifolds). Our result covers, in particular, the case of 0-framed surgeries
along knots of genus one. The statement follows from the proof that twisted
Alexander polynomials decide fiberability for all the 3-manifolds listed above.
As a corollary, it follows that twisted Alexander polynomials decide if a knot
of genus one is fibered.
