On the Sigma invariants of even Artin groups of FC-type
In this paper we study Sigma invariants of even Artin groups of FC-type, extending some known results for right-angled Artin groups. In particular, we define a condition that we call the strong homological $n$-link condition for a graph $Γ$ and prove that it gives a sufficient condition for a character $χ:A_Γ\to \mathbb{Z}$ to satisfy $[χ]\inΣ^n(A_Γ,\mathbb{Z})$. This implies that the kernel $A^χ_Γ=\ker χ$ is of type $FP_n$. The homotopy counterpart is also proved. Partial results on the converse are discussed. We also provide a general formula for the free part of $H_n(A^χ_Γ;\mathbb{F})$ as an $\mathbb{F}[t^{\pm 1}]$-module with the natural action induced by $χ$. This gives a characterization of when $H_n(A^χ_Γ;\mathbb{F})$ is a finite dimensional vector space over $\mathbb{F}$. In the last version we correct a problem in the proof of Lemma 4.3 and also a remark at the end of subsection 3.3.