On the Bieri-Neumann-Strebel-Renz $\Sigma^1$-invariant of even Artin groups

Abstract

We calculate the Bieri-Neumann-Strebel-Renz invariant $\Sigma^1(G)$ for even Artin groups $G$ with underlying graph $\Gamma$ such that if there is a closed reduced path in $\Gamma$ with all labels bigger than 2 then the length of such path is always odd. We show that $\Sigma^1(G)^c$ is a rationally defined spherical polyhedron.