We investigate the behavior of the higher-order degrees, $\bar{\delta}_n$, of a finitely presented group $G$. These $\bar{\delta}_n$ are functions from $H^1(G;{\mathbb Z})$ to ${\mathbb Z}$ whose values are the degrees certain higher-order Alexander polynomials. We show that if ${\rm def}(G)\geq 1$ or $G$ is the fundamental group of a compact, orientable 3-manifold then $\bar{\delta}_n$ is a monotonically increasing function of $n$ for $n\geq1$. 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 previous results of the author. We also find new conditions on a 3-manifold $X$ that will guarantee that the Thurston norm of $f^{\ast}(\psi)$, for $\psi \in H^1(X;{\mathbb Z})$ and $f{:}\,Y \rightarrow X$ 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, $\bar{\delta}_\Gamma$, corresponding to a surjective map $G\twoheadrightarrow \Gamma$ for which $\Gamma$ is poly-torsion-free-abelian. Under certain conditions, we show they satisfy a monotonicity condition if one varies the group $\Gamma$. 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.