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

TL;DR

This paper computes the Bieri-Neumann-Strebel-Renz invariant for a class of even Artin groups, revealing that the complement of this invariant forms a rational spherical polyhedron under certain graph conditions.

Contribution

It provides a precise calculation of the $ ext{Sigma}^1$-invariant for specific even Artin groups based on their defining graph properties.

Findings

$ ext{Sigma}^1(G)^c$ is a rational spherical polyhedron.

The invariant is explicitly determined for groups with certain graph conditions.

The structure of the invariant depends on the parity of the length of closed paths in the graph.

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.