Alexander invariants and cohomology jump loci in group extensions

TL;DR

This paper investigates the relationships between Alexander invariants, cohomology jump loci, and group extension structures, revealing precise connections and inequalities, especially in split extensions, across integral, rational, and modular settings.

Contribution

It establishes a detailed relationship between Alexander invariants and cohomology jump loci in group extensions with trivial monodromy, including new inequalities and equalities for Chen ranks.

Findings

Relationship between Alexander invariants and characteristic/ resonance varieties

Inequality between Chen ranks in group extensions

Equality of Chen ranks in degrees > 1 for split extensions

Abstract

We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. 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 $H_1(K;\mathbb{Z})$, with suitable modifications in the rational and mod-$p$ settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups $K$ and $G$. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.