  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.
