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

TL;DR

The paper studies when a Bestvina--Brady group $BB_{\,\Gamma}$ is a RAAG, providing a graph-theoretic criterion based on a tree $2$-spanner that yields a RAAG presentation $BB_{\,\Gamma}\cong A_{T^{*}}$ and showing that the class of BBGs contains all RAAGs. It develops a graphical description of the BNS-invariant $\Sigma^{1}(BB_{\, obreak ext Gamma})$, using dead-edge subgraphs and edge-labellings to give explicit criteria for (non)membership, and connects these invariants to resonance varieties, enabling obstruction results beyond existing criteria. In dimension two, the authors prove an equivalence: $BB_{\, obreak extGamma}$ is a RAAG if and only if $ obreak\Delta_{ obreak extGamma}$ has no crowned triangles, which is further equivalent to the existence of a tree $2$-spanner for $ obreak extGamma$, thereby solving the RAAG-recognition problem in this setting. The work also develops criteria based on redundant triangles to certify non-RAAG BBGs and shows how crown- and trefoil-type configurations influence whether a BBG can be an Artin group, with resonance theory providing a deeper structural understanding of these obstructions.

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.

Figures (20)

• Figure 1: The trefoil graph
• Figure 2: The extended trefoil graph. The BBG defined by it has this presentation: $\langle a,b,c,d,e,f \ \vert \ [a,b], [b,c], [c,d], [b^{-1}c,e], [e,f] \rangle$
• Figure 3: The implications in the proof of Theorem \ref{['main thm 2dim']}
• Figure 4: Oriented triangle.
• Figure 5: The construction of the loop $L$ from the cycle $C$ (left), and its contraction to a cone vertex (right).

Theorems & Definitions (129)

• Theorem A
• Theorem B
• Corollary 1
• Theorem C: Graphical criterion for the BNS-invariant of a BBG
• Theorem D: Graphical description of the BNS-invariant of a BBG
• Theorem E
• Theorem 2.1
• Corollary 2.2
• Remark 2.3
• Example 2.4