On the $Σ$-invariants of Artin groups satisfying the $K(π,1)$-conjecture

Abstract

We consider $Σ$-invariants of Artin groups that satisfy the $K(π,1)$-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 $χ:A_Γ\to\mathbb{R}$ to belong to $Σ^n(A_Γ,\mathbb{Z})$. We also prove some partial converses. As applications, we prove that the $Σ^1$-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 $Σ$-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.

Theorems & Definitions (79)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Bieri-Renz Bieri-Renz
• Lemma 2.4: Meier-Meiner-VanWyk 2 Lemma 2.1
• Theorem 2.5: Meier-Meiner-VanWyk Theorem 3.2