Summary: For a 3-manifold \(M\), C. T. McMullen [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, 153–171 (2002; Zbl 1009.57021)] derived from the Alexander polynomial of \(M\) a norm on \(H^1(M,\mathbb{R})\) called the Alexander norm. He showed that the Thurston norm is an upper bound for the Alexander norm. He asked whether these two norms are 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.