Alexander invariants and cohomology jump loci in group 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.

arXiv comment62 pages; accepted for publication in Annali della Scuola Normale Superiore di Pisa

References in corpus (18)

• Algebraic invariants for right-angled Artin groups
• Fundamental groups of line arrangements: Enumerative aspects
• Chen Lie algebras
• Toric complexes and Artin kernels
• Algebraic invariants for Bestvina-Brady groups
• Formality, Alexander invariants, and a question of Serre
• Geometric and algebraic aspects of 1-formality
• Formality properties of finitely generated groups and Lie algebras
• Homology and derived series of groups
• On finite generation of the Johnson filtrations
• Homotopy types of complements of 2-arrangements in R^4
• Alexander Invariants of Complex Hyperplane Arrangements
• Chen ranks and resonance
• New lower bounds on subgroup growth and homology growth
• Cup products, lower central series, and holonomy Lie algebras
• Chen ranks and resonance varieties of the upper McCool groups
• Exact sequences, lower central series and representations of surface braid groups
• Lower central series and split extensions

Cited by in corpus (1)

• Milnor fibrations of arrangements with trivial algebraic monodromy