  For a 3-manifold M, McMullen derived from the Alexander polynomial of M a
norm on H^1(M, R) called the Alexander norm. He showed that the Thurston norm
on H^1(M, R), which measures the complexity of a dual surface, is an upper
bound for the Alexander norm. He asked if these two norms were equal on all of
H^1(M,R) when M fibers over the circle. Here, I give examples which show that
the answer to this question is emphatically no. This question is related to the
faithfulness of the Gassner representations of the braid groups. The key tool
used to understand this question is the Bieri-Neumann-Strebel invariant from
combinatorial group theory. Theorem 1.7, which is of independant interest,
connects the Alexander polynomial with a certain Bieri-Neumann-Strebel
invariant.
