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

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 tree, 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 with a 2-dimensional flag complex is a RAAG in terms of certain subgraphs.