On the $\Sigma$-invariants of Artin groups satisfying the $K(\pi,1)$-conjecture

TL;DR

This paper investigates the $ ext{Sigma}$-invariants of Artin groups satisfying the $K( extpi,1)$-conjecture, extending known results, proving new conjectures, and classifying invariants for various Artin groups.

Contribution

It extends results for even Artin groups of FC-type, provides conditions for $ ext{Sigma}$-invariants, and classifies invariants for spherical, affine, and triangle Artin groups.

Findings

Proves the $ ext{Sigma}^1$-conjecture under specific divisibility conditions.

Generalizes the homological Bestvina-Brady theorem to Artin groups.

Computes $ ext{Sigma}$-invariants for all irreducible spherical, affine, and triangle Artin groups.

Abstract

We consider $\Sigma$-invariants of Artin groups that satisfy the $K(\pi,1)$-conjecture. These invariants determine the cohomological finiteness conditions of subgroups that contain the derived subgroup. We extend a known result for even Artin groups of FC-type, giving a sufficient condition for a character $\chi:A_\Gamma\to\mathbb{R}$ to belong to $\Sigma^n(A_\Gamma,\mathbb{Z})$. We also prove some partial converses. As applications, we prove that the $\Sigma^1$-conjecture holds true when there is a prime $p$ that divides $l(e)/2$ for any edge with even label $l(e)>2$, we generalize to Artin groups the homological version of Bestvina-Brady theorem and we compute the $\Sigma$-invariants of all irreducible spherical and affine Artin groups and triangle Artin groups, which provide a complete classification of the $F_n$ and $FP_n$ properties of their derived subgroup.