Pro-C RAAGs

Abstract

Let $C$ be a class of finite groups closed under taking subgroups, quotients, and extensions with abelian kernel. The right-angled Artin pro-$C$ group $G_{\Gamma }$ (pro-$C$ RAAG for short) is the pro-$C$ completion of the right-angled Artin group $G\left(\Gamma \right)$ associated with the finite simplicial graph Γ.

In the first part, we describe structural properties of pro-$C$ RAAGs. Among others, we describe the centraliser of an element and show that pro-$C$ RAAGs satisfy the Tits' alternative, that standard subgroups are isolated, and that 2-generated pro-p subgroups of pro-$C$ RAAGs are either free pro-p or free abelian pro-p.

In the second part, we characterise splittings of pro-$C$ RAAGs in terms of the defining graph. More precisely, we prove that a pro-$C$ RAAG $G_{\Gamma }$ splits as a non-trivial direct product if and only if Γ is a join and it splits over an abelian pro-$C$ group if and only if a connected component of Γ 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-$C$ RAAG, in the sense of Guirardel and Levitt [9].