Monotonicity of Degrees of Generalized Alexander Polynomials of Groups and 3-Manifolds

Abstract

We investigate the behavior of the higher-order degrees, db_n, of a finitely presented group G. These db_n are functions from H^1(G;Z) to Z whose values are the degrees certain higher-order Alexander polynomials. We show that if def(G) is at least 1 or G is the fundamental group of a compact, orientable 3-manifold then db_n is a monotonically increasing function of n for n at least 1. This is false for general groups. As a consequence, we show that if a 4 manifold of the form X times S^1 admits a symplectic structure then X ``looks algebraically like'' a 3-manifold that fibers over S^1, supporting a positive answer to a question of Taubes. This generalizes a theorem of S. Vidussi and is an improvement on the previous results of the author. We also find new conditions on a 3-manifold X which will guarantee that the Thurston norm of f*(psi), for psi in H^1(X;\Z) and f:Y -> X a surjective map on pi_1, will be at least as large the Thurston norm of psi. When X and Y are knot complements, this gives a partial answer to a question of J. Simon. More generally, we define Gamma-degrees, db_Gamma, corresponding to a surjective map G -> Gamma for which Gamma is poly-torsion-free-abelian. Under certain conditions, we show they satisfy a monotonicity condition if one varies the group. As a result, we show that these generalized degrees give obstructions to the deficiency of a group being positive and obstructions to a finitely presented group being the fundamental group of a compact, orientable 3-manifold.