Finite graph product closure for a conjecture on the BNS-invariant of Artin groups
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Archiv der Mathematik
Finite graph product closure for a conjecture on the BNS-invariant of Artin groups Kisnney Emiliano de Almeida
and Francismar Ferreira Lima
Abstract. We work with a conjecture on the BNS-invariant of Artin groups stated by Almeida and Kochloukova. We show that the class of Artin groups that satisfy this conjecture is closed under finite graph products. As a consequence, we show that the conjecture is true for all Artin groups of finite type and other subclasses. Mathematics Subject Classification. 20F36, 20F05. Keywords. Sigma-invariants, Artin groups, BNS-invariant.
1. Introduction. The Bieri-Neumann-Strebel invariant, also called Σ1 invariant, was first defined in [6] and has a lot of equivalent definitions, one of which we present below. Let G be a finitely generated group and S(G) be the sphere of characters , where two homomorphisms χ1 , χ2 are equivalent of G, given by Hom(G,R)\{0} ∼ if χ1 = rχ2 for some r ∈ R+ . Let Γ be a Cayley graph of G relative to any finite generating set and Γχ be the full subgraph of Γ generated by Gχ := {g ∈ G | χ(g) ≥ 0}. Then, by definition, Σ1 (G) := {[χ] ∈ S(G) | Γχ is connected}. The main application of this invariant is given by Bieri-Neumann-Strebel’s result [6] which states that if G is a finitely generated group and H is a subgroup of G containing G , then H is finitely generated if and only if the classes of characters [χ] of G for which χ(H) = 0 are all inside Σ1 (G). Bieri and Renz [7] generalized the BNS-invariant and defined descending chains of topological and homological invariants Σn for n ≥ 2 that gave rise to a similar and much deeper result. The definition of those invariants is quite technical, but the result is a natural generalization of the original Bieri-NeumannStrebel theorem: If G is a group of type FPm (resp. Fm ) and H is a subgroup of
K.E. de Almeida and F.F. Lima
Arch. Math.
G containing G , then H is of type FPm (resp. Fm ) if and only if the classes of characters [χ] of G for which χ(H) = 0 are all inside Σm (G, Z) (resp. Σm (G)). Those Σ-invariants are usually very hard to compute and a lot of effort has been done to accomplish it for several families of groups. The first step towards this computing usually is to find Σ1 since it has a much simpler definition, many more avaliable results, and contains all the other Σ-invariants. For example, Zaremsky [14] has obtained Σn of the generalized Thompson groups Fm,∞ for all n, m. This was the final step of a series of hard previous calculations by many authors, first Σ1 (F2,∞ ) [6], then Σn (F2,∞ ) for all n [5], then Σ2 (Fm,∞ ) for all m [8], and finally Zaremsky’s general result. Artin groups are finitely presented groups combinatorially defined from a labeled simplicial graph. Meier, Meinert, and VanWyk [11] have obtained a necessary and a sufficient condition suitable for all Artin groups for a character to be in Σ1 , depending on the topology of the group’s underlying graph and two of its subgraphs, L(χ) and LF (χ). Meier [9] has proven that the above sufficient condition is also necessary for
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