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

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 $F{P}_n$ properties of their derived subgroup.

25 pages. Revised version, some missprints corrected. Accepted for publication in the JLMS

References in corpus (3)

• Cohomology of Artin groups of type tilde{A}_n, B_n and applications
• On the Bieri-Neumann-Strebel-Renz ${\Sigma }^1$-invariant of even Artin groups
• Homology of even Artin kernels

Cited by in corpus (1)

• Around subgroups of Artin groups: derived subgroups and acylindrical hyperbolicity in the even FC-case