2023/11/22 by Casals-Ruiz, Montserrat, Pintonello, Matteo, Zalesskii, Pavel
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2311.13439
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 GΓ (pro-C RAAG for short) is the pro-C completion of the right-angled Artin group G(Γ) 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 GΓ 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.