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

Abstract

Let $C$ be a class of finite groups closed for subgroups, quotient groups and extensions. Let Γ be a finite simplicial graph and $G=G_{\Gamma }$ 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 $F{P}_n$ when $G/N\simeq Z_p$. Furthermore for $G/N$ infinite abelian we show that N is finitely generated if and only if every normal closed subgroup $N_0◃G$ containing N with $G/N_0\simeq Z_p$ is finitely generated. For $G/N$ infinite abelian with N weakly discretely embedded in G we show that N is of type $F{P}_n$ if and only if every $N_0⩽G$ containing N with $G/N_0\simeq Z_p$ is of type $F{P}_n$.