Profinite properties of RAAGs and special groups

TL;DR

This paper demonstrates that right-angled Artin groups (RAAGs) and right-angled Coxeter groups (RACGs) can be distinguished by their pro-$p$ and pro-2$ completions respectively, and explores properties linking graphs to profinite completions.

Contribution

It establishes the uniqueness of pro-$p$ and pro-2$ completions for RAAGs and RACGs, respectively, and provides a new proof of goodness for hyperbolic virtually special groups.

Findings

RAAGs are distinguished by their pro-$p$ completions for any prime $p$

RACGs are distinguished by their pro-2$ completions

Hyperbolic virtually special groups are good in the sense of Serre

Abstract

In this paper we prove that RAAGs are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion.