2020/10/07 by Dessislava H. Kochloukova, Luís Mendonça, Kochloukova, Dessislava H. +2
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR
paper · pdf · doi:10.48550/arxiv.2010.03689
arxiv created 2020/10/07 · openalex publication_date 2020/10/07 · arxiv updated 2020/10/09 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We study the Bieri-Neumann-Strebel-Renz invariants and we prove the following criterion: for groups H and K of type FPn such that [H,H] ⊆ K ⊆ H and a character χ: K → ℝ with χ([H,H]) = 0 we have [χ] ∈ Σn(K, ℤ) if and only if [μ] ∈ Σn(H, ℤ) for every character μ: H → ℝ that extends χ. The same holds for the homotopical invariants Σn(-) when K and H are groups of type Fn. We use these criteria to complete the description of the Σ-invariants of the Bieri-Stallings groups Gm and more generally to describe the Σ-invariants of the Bestvina-Brady groups. We also show that the "only if" direction of such criterion holds if we assume only that K is a subnormal subgroup of H, where both groups are of type FPn. We apply this last result to wreath products.