Alexander and Thurston norms of fibered 3-manifolds

Details Alexander and Thurston norms of fibered 3-manifolds Alexander and Thurston norms of fibered 3-manifolds

1999-08-11

arxiv.org/abs/math/9908050v2

Pacfic J. Math, 200 (2001), 43-58

Mathematics

Geometric Topology

14 pages, 7 figures. To appear in PJM. v2: Minor corrections over v1

Scientific paper

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.

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