On the Sigma invariants of even Artin groups of FC-type

Abstract

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 $\Gamma$ and prove that it gives a sufficient condition for a character $\chi :A_{\Gamma }\to \mathbb{Z}$ to satisfy $[\chi ]\in {\Sigma }^n(A_{\Gamma },\mathbb{Z})$. This implies that the kernel $A_{\Gamma }^{\chi }=\ker \chi$ 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 $H_n(A_{\Gamma }^{\chi };\mathbb{F})$ as an $\mathbb{F}[t^{\pm 1}]$-module with the natural action induced by $\chi$. This gives a characterization of when $H_n(A_{\Gamma }^{\chi };\mathbb{F})$ 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.

19 pages