Finitely generated normal pro-[formula omitted] subgroups in right angled Artin pro-[formula omitted] groups

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=GΓ be the corresponding pro-C RAAG. We show that if N is a non-trivial finitely generated, normal, full pro-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 FPn when G/N≃Zp. Furthermore for G/N infinite abelian we show that N is finitely generated if and only if every normal closed subgroup N0◃G containing N with G/N0≃Zp is finitely generated. For G/N infinite abelian with N weakly discretely embedded in G we show that N is of type FPn if and only if every N0⩽G containing N with G/N0≃Zp is of type FPn.