Right Angled Artin Groups and partial commutation, old and new

Abstract

We systematically treat algebraic objects with free partially commuting generators and give short and modern proofs of the various relations between them. These objects include right angled Artin groups, polynomial rings, Lie algebras, and restricted Lie algebras in partially commuting free generators. In particular, we compute the $p$-central and exponent-$p$ series of all right angled Artin groups, and compute the dimensions of their subquotients. We also describe their associated Lie algebras, and relate them to the cohomology ring of the group as well as to polynomial and power series rings in partially commuting variables. We finally show how the growth series of these various objects are related to each other.