Alexander and Thurston norms of fibered 3-manifolds

For a 3-manifold M, McMullen derived from the Alexander polynomial of M a norm on H¹ (M, ℝ) called the Alexander norm. He showed that the Thurston norm is an upper bound for the Alexander norm. He asked if these two norms were the same when M fibers over the circle. Here, I give examples that show this is not the case. This question relates to the faithfulness of the Gassner representations of the braid groups. The key tool used is the Bieri-Neumann-Strebel invariant, and I show a connection between this invariant and the Alexander polynomial.