A graphical description of the BNS-invariants of Bestvina-Brady groups and the RAAG recognition problem

TL;DR

This paper characterizes when Bestvina-Brady groups are right-angled Artin groups using graph-theoretic criteria and Bieri-Neumann-Strebel invariants, providing tools to distinguish BBGs from RAAGs.

Contribution

It introduces a graph-based criterion for BBGs to be RAAGs and offers a method to identify when BBGs are not RAAGs, expanding understanding of their structure.

Findings

Certain spanning trees imply BBGs are RAAGs

A criterion to distinguish non-RAAG BBGs

Characterization of RAAGs from 2-dimensional flag complexes

Abstract

A finitely presented Bestvina-Brady group (BBG) admits a presentation involving only commutators. We show that if a graph admits a certain type of spanning trees, then the associated BBG is a right-angled Artin group (RAAG). As an application, we obtain that the class of BBGs contains the class of RAAGs. On the other hand, we provide a criterion to certify that certain finitely presented BBGs are not isomorphic to RAAGs (or more general Artin groups). This is based on a description of the Bieri-Neumann-Strebel invariants of finitely presented BBGs in terms of separating subgraphs, analogous to the case of RAAGs. As an application, we characterize when the BBG associated to a 2-dimensional flag complex is a RAAG in terms of certain subgraphs.