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We study the integral, rational, and modular Alexander invariants, as well asthe cohomology jump loci of groups arising as extensions with trivial algebraicmonodromy. Our focus is on extensions of the form $1\to K\to G\to Q\to 1$,where $Q$ is an abelian group acting trivially on $H_1(K;\mathbb{Z})$, withsuitable modifications in the rational and mod-$p$ settings. We find a tightrelationship between the Alexander invariants, the characteristic varieties,and the resonance varieties of the groups $K$ and $G$. This leads to aninequality between the respective Chen ranks, which becomes an equality indegrees greater than 1 for split extensions.

Alexander invariants and cohomology jump loci in group extensions