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

TL;DR

This paper investigates Sigma invariants of even Artin groups of FC-type, establishing conditions for their homological properties and providing formulas for their homology groups, extending known results from right-angled Artin groups.

Contribution

It introduces the strong homological n-link condition for graphs, linking it to Sigma invariants and kernel finiteness properties in even Artin groups of FC-type.

Findings

The strong homological n-link condition ensures membership in Sigma^n.

Provides a formula for the free part of homology groups as modules.

Characterizes when homology groups are finite dimensional over a field.

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 $A^\chi_\Gamma=\ker \chi$ 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 $H_n(A^\chi_\Gamma;\mathbb{F})$ as an $\mathbb{F}[t^{\pm 1}]$-module with the natural action induced by $\chi$. This gives a characterization of when $H_n(A^\chi_\Gamma;\mathbb{F})$ 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.