On the Sigma invariants of even Artin groups of FC-type

2020/11/15 by Ruben Blasco.García, García, Ruben Blasco., José Agustín +3 · 1 citation
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2011.07608

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 homological 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]∈Σⁿ(A_Γ,ℤ). This implies that the kernel A^χ_Γ=ker χ is of type FP_n. The homotopy counterpart is also proved. Partial results on the converse are discussed. We also provide a general formula for the free part of H_n(A^χ_Γ;\mathbbF) as an \mathbbF[t^{± 1}]-module with the natural action induced by χ. This gives a characterization of when H_n(A^χ_Γ;\mathbbF) is a finite dimensional vector space over \mathbbF. In the last version we correct a problem in the proof of Lemma 4.3 and also a remark at the end of subsection 3.3.

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