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

Abstract

Let $\mathcal{C}$ be a class of finite groups closed for subgroups, quotients groups and extensions. Let $\Gamma$ be a finite simplicial graph and $G = G_{\Gamma}$ 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 $G/ N \simeq \mathbb{Z}_p$. Furthermore for $G/ N$ infinite abelian we show that $N$ is finitely generated if and only if every normal closed subgroup $ N_0 \triangleleft G$ containing $N$ with $G/ N_0 \simeq \mathbb{Z}_p$ 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 $ N_0 \leq G$ containing $N$ with $G/ N_0 \simeq \mathbb{Z}_p$ is of type $FP_n$.