The BNS-invariant for Artin groups of circuit rank 2
The Artin group \(G\) with underlying finite simplicial graph \(\mathcal{G}\), with edges labeled by integers greater than one, is given by a finite presentation, with generators corresponding to the vertices \(V(\mathcal{G})\) of \(\mathcal{G}\) and relations given by \([u, v]_n = [v, u]_n\) for each edge \(\{u, v \}\) of \(\mathcal{G}\) with label \(n\), where \[ [a, b]_n = \underbrace{a b a b \ldots}_{n \text{ factors}} \text{ for all }a, b \in V(\mathcal{G}). \] If \(\mathcal{G}\) is connected and \(\pi_1 (\mathcal{G})\) is free of rank \(n\), then \(G\) is called an Artin group of \textit{circuit rank} \(n\). Let \(\mathcal{G}\) be a graph. A subgraph \(\mathcal{H}\) of \(\mathcal{G}\) is called \textit{dominant} if for each vertex \(v \in V(\mathcal{G}) \setminus V(\mathcal{H})\) there is an edge \(e \in E(\mathcal{G})\) such that \(\sigma(e) = v\) and \(\tau(e) \in V(\mathcal{H})\). Let \(\chi\) be a character of an Artin group \(G\) with underlying graph \(\mathcal{G}\). An edge \(e\) of \(\mathcal{G}\) is called \textit{dead} if \(e\) has an even label greater than 2 and \(\chi(\sigma(e)) = -\chi(\tau(e))\). Let \(G\) be an Artin group with underlying graph \(\mathcal{G}\) and let \(\chi : G \to \mathbb{R}\) be a non-zero real character of \(G\). Define \(\mathcal{L}_{\mathcal{F}} = \mathcal{L}_{\mathcal{F}}(\chi)\) as the full subgraph of \(\mathcal{G}\) generated by the vertices \(v \in V(\mathcal{G})\) such as \(\chi(v) \not= 0\). Define the \textit{living subgraph} \(\mathcal{L} = \mathcal{L}(\chi) \subset \mathcal{L}_{\mathcal{F}}\) as the subgraph obtained from \(\mathcal{L}_{\mathcal{F}}\) after removing the dead edges. The BNS-invariant or \(\Sigma^1\)-invariant is the first of a series of geometric invariants of finitely generated groups defined in the eighties that are deeply related to finiteness properties of their subgroups. \textit{J. Meier} [Proc. Lond. Math. Soc. (3) 74, No. 1, 151--173 (1997; Zbl 0869.20026)] calculated \(\Sigma^1(G)\) when \(\mathcal{G}\) is a tree and in [\textit{J. Meier} et al., Topology Appl. 110, No. 1, 71--81 (2001; Zbl 1037.20037)], some sufficient and necessary conditions were obtained for an element \([\chi]\) of the unit sphere \(S(G)\) to be in \(\Sigma^1(G)\). In [Forum Math. 27, No. 5, 2901--2925 (2015; Zbl 1362.20031)], the author and \textit{D. Kochloukova} showed that the sufficient condition of [Meier et al., loc. cit.] is also necessary for Artin groups of circuit rank 1 and some other subclasses. In the present paper, the same result for Artin groups of circuit rank 2 is shown. More accurately, it is proved that if \(G\) is an Artin group of circuit rank 2, with underlying graph \(\mathcal{G}\), then \[ \Sigma^1(G) = \{ [\chi] \in S(G) \mid \mathcal{L}(\chi) \text{ is a connected dominant subgraph of } \mathcal{G}\}. \]
scientific article; zbMATH DE number 6850106