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

Abstract

We consider ‐invariants of Artin groups that satisfy the ‐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 to belong to . We also prove some partial converses. As applications, we prove that the ‐conjecture holds true when there is a prime that divides for any edge with even label , we generalize to Artin groups the homological version of the Bestvina–Brady theorem and we compute the ‐invariants of all irreducible spherical and affine Artin groups and triangle Artin groups, which provide a complete classification of the and properties of their derived subgroup.