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→K→G→Q→1, where Q is an abelian group acting trivially on H1(K;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.

Lower central series and split extensions

“…By our assumption, if c P C 1 and x P A 1 , then x c " rc ´1, xsx P A 1 . Thus, if a P A 1 , then xrc, asx ´1 belongs to rA 1 , C 1 s, by formula (34). The claim and its consequences follow at once.…”

“…Clearly, if C acts trivially on A abf , then it acts trivially on A abf b Q. As we note in [34], the reverse implication holds if A abf is finitely generated, but it is not valid in general.…”

“…We pursue this investigation in [34], where we study the derived series, G Ě G 1 Ě G 2 Ě ¨¨¨, and the Alexander invariant, BpGq " G 1 {G 2 , as well as the rational and pversions of these objects, together with the cohomology jump loci of a finitely generated group G. The framework developed here, and the results herein, are used in an essential way in [34], as well as in a planned sequel, [35], where we apply these techniques to the study of Milnor fibrations of hyperplane arrangements. 1.7.…”