The BNS-invariant for Artin groups of circuit rank 2

Abstract

Abstract The BNS-invariant or Σ 1 {\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, although they are very hard to compute. Meier, Meinert and VanWyk have obtained a partial description of Σ 1 {\Sigma^{1}} of Artin groups, but the complete description of the general case is still an open problem. Let the circuit rank of an Artin group be the free rank of the fundamental group of its underlying graph. Meier, in a previous work, obtained a complete description for Artin groups of circuit rank 0, i.e., whose underlying graphs are trees. In a previous work we have proved, in joint work with Kochloukova, the same description to be true for Artin groups of circuit rank 1. In this paper we prove the description to be true for every Artin group of circuit rank 2.