The $\Sigma^1$ invariant for some Artin groups of arbitrary circuit rank

TL;DR

This paper classifies the Bieri-Neumann-Strebel-Renz invariant for certain Artin groups with complex graph structures, advancing understanding of their algebraic and geometric properties.

Contribution

It provides a classification of the $\Sigma^1$ invariant for a new class of Artin groups with arbitrary circuit rank, extending previous results.

Findings

Complete classification of $\Sigma^1$ invariant for these Artin groups

Identification of conditions affecting the invariant's structure

Extension of known results to more complex graph configurations

Abstract

We classify the Bieri-Neumann-Strebel-Renz invariant $\Sigma^1(G)$ for a class of Artin groups with minimal graphs of arbitrary circuit rank.