On the $\Sigma^1$ and $\Sigma^2$-invariants of Artin groups

TL;DR

This paper advances understanding of the $\Sigma^1$ and $\Sigma^2$ invariants in Artin groups by proving conjectures for specific families, including balanced and 2-dimensional Artin groups, and computing invariants for all dimensions in the 2D case.

Contribution

It proves the $\Sigma^1$-conjecture for new families of Artin groups and confirms a conjecture on the $\Sigma^2$-invariant for groups satisfying the $K(\pi,1)$-conjecture, including explicit computations.

Findings

Proved the $\Sigma^1$-conjecture for certain Artin groups.

Confirmed the $\Sigma^2$-conjecture for 2-dimensional and coherent Artin groups.

Computed $\Sigma^n$ for all $n extgreater{}2$ in the 2D case.

Abstract

We prove the $\Sigma^1$-conjecture for two families of Artin groups: Artin groups such that there exists a prime number $p$ dividing $\frac{l(e)}{2}$ for every edge $e$ with even label $>2$ and balanced Artin groups. The family of balanced Artin groups extends two previously studied families: the one considered by Kochloukova in arXiv:2009.14269, and the family of coherent Artin groups. We state a conjecture on the $\Sigma^2$-invariant for Artin groups satisfying the $K(\pi,1)$-conjecture. The conjecture is proven to be true for two significant families: $2$-dimensional and coherent Artin groups. In the $2$-dimensional case we are able to compute $\Sigma^n$ for all $n\geq 2$ and to derive finiteness properties of the derived subgroup.