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 $\mathrm{\Gamma }$ and prove that it gives a sufficient condition for a character $\chi :A_{\mathrm{\Gamma }}\to \mathbb{Z}$ to satisfy $[\chi ]\in {\mathrm{\Sigma }}^n(A_{\mathrm{\Gamma }},\mathbb{Z})$. This implies that the kernel $A_{\mathrm{\Gamma }}^{\chi }=\ker \chi$ is of type $F{P}_n$. The homotopy counterpart is also proved. Partial results on the converse are discussed. We also provide a general formula for the free part of $H_n(A_{\mathrm{\Gamma }}^{\chi };\mathbb{F})$ as an $\mathbb{F}[t^{\pm 1}]$-module with the natural action induced by $\chi$. This gives a characterization of when $H_n(A_{\mathrm{\Gamma }}^{\chi };\mathbb{F})$ 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.