Finitely generated normal pro-\mathcal C subgroups in right angled Artin pro-\mathcal C groups

2023/05/05 by Dessislava H. Kochloukova, Kochloukova, Dessislava, Pavel Zalesskii +1
Mathematics · #Geometric and Algebraic Topology #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2305.03683

Abstract

Let C be a class of finite groups closed for subgroups, quotients groups and extensions. Let Γ be a finite simplicial graph and G = G_Γ be the corresponding pro-\mathcal C RAAG. We show that if N is a non-trivial finitely generated, normal, full pro-\mathcal C subgroup of G then G/ N is finite-by-abelian. In the pro-p case we show a criterion for N to be of type FP_n when G/ N ≃ ℤ_p. Furthermore for G/ N infinite abelian we show that N is finitely generated if and only if every normal closed subgroup N₀ \triangleleft G containing N with G/ N₀ ≃ ℤ_p is finitely generated. For G/ N infinite abelian with N weakly discretely embedded in G we show that N is of type FP_n if and only if every N₀ ≤ G containing N with G/ N₀ ≃ ℤ_p is of type FP_n.

Related

• Pro-C RAAGs
• On finitely generated normal subgroups of right-angled Artin groups and graph products of groups
• Right-angled Artin pro-p groups
• Subgroup properties of pro-p extensions of centralizers
• Subgroups of Bestvina-Brady groups