Abstract
In this paper we study Sigma-invariants of even Artin groups of FC-type, extending some known results for right-angled Artin groups. In particular, we define a condition that we call the strong n -link condition for a graph Γ and prove that it gives a sufficient condition for a character χ : A Γ → Z to satisfy [ χ ] ∈ Σ n ( A Γ , Z ) . This implies that the kernel A Γ χ = ker χ is of type FP n . We prove the homotopical version of this result as well and discuss partial results on the converse. We also provide a general formula for the free part of H n ( A Γ χ ; F ) as an F [ t ± 1 ] -module with the natural action induced by χ . This gives a characterization of when H n ( A Γ χ ; F ) is a finite dimensional vector space over F .
Highlights
The so-called Sigma invariants of a group G are certain sets Σn(G, Z), Σn(G) of equivalence classes of characters χ : G → R that provide information about the cohomological – in the case of Σn(G, Z) – and homotopical – for Σn(G) – finiteness conditions of subgroups lying over the commutator of G
It is extremely difficult to compute these invariants explicitly but there are some remarkable cases in which a full computation is available. One of those cases occurs when G is a right-angled Artin group (RAAG for short). These groups are defined from a given finite graph Γ which will be assumed here to be simple, i.e., without loops or multiple edges between vertices
Associated to Γ one can describe the RAAG, denoted by AΓ, as the group generated by the vertices of Γ with relators of the form [v, w] = 1 for any edge {v, w} of Γ
Summary
The so-called Sigma invariants of a group G are certain sets Σn(G, Z), Σn(G) of equivalence classes of characters χ : G → R that provide information about the cohomological – in the case of Σn(G, Z) – and homotopical – for Σn(G) – finiteness conditions of subgroups lying over the commutator of G. Given a finite simple graph Γ as above, one can consider an even labeling on the edges, that is, for any edge e = {u, v}, its label l(e) is an even number Any such even graph Γ defines an even Artin group AΓ generated by the vertices of Γ and whose relators have the form (uv)k = (vu)k, where l(e) = 2k. Let G = AΓ be an even Artin group of FC-type, and 0 = χ : G → R a character such that the strong n-link condition holds for χ. For n = 1 it is known for several types of Artin groups (see Theorem 3.6 in Subsection 3.2) that [χ] ∈ Σn(G, Z) if and only if Lχ is connected and dominant This is equivalent to saying that the strong n-link condition holds for χ (see Subsection 3.1).
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call