Pro-$\mathcal{C}$ RAAGs

Abstract

Let $\mathcal{C}$ be a class of finite groups closed under taking subgroups, quotients, and extensions with abelian kernel. The right-angled Artin pro-$\mathcal{C}$ group $G_\Gamma$ (pro-$\mathcal{C}$ RAAG for short) is the pro-$\mathcal{C}$ completion of the right-angled Artin group $G(\Gamma)$ associated with the finite simplicial graph $\Gamma$. In the first part, we describe structural properties of pro-$\mathcal{C}$ RAAGs. Among others, we describe the centraliser of an element and show that pro-$\mathcal{C}$ RAAGs satisfy the Tits' alternative, that standard subgroups are isolated, and that 2-generated pro-$p$ subgroups of pro-$\mathcal{C}$ RAAGs are either free pro-$p$ or free abelian pro-$p$. In the second part, we characterise splittings of pro-$\mathcal{C}$ RAAGs in terms of the defining graph. More precisely, we prove that a pro-$\mathcal{C}$ RAAG $G_\Gamma$ splits as a non-trivial direct product if and only if $\Gamma$ is a join and it splits over an abelian pro-$\mathcal{C}$ group if and only if a connected component of $\Gamma$ is a complete graph or it has a complete disconnecting subgraph. We then use this characterisation to describe an abelian JSJ decomposition of a pro-$\mathcal{C}$ RAAG, in the sense of Guirardel and Levitt.