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Symplectic $ \mathbf{S}^{1} \times N^3$, subgroup separability, and vanishing Thurston norm
Authors:Stefan Friedl  Stefano Vidussi
Institution:Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada ; Department of Mathematics, University of California, Riverside, California 92521
Abstract: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.

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