# Right Angled Artin Groups and partial commutation, old and new

## TL;DR

This paper analyzes the algebraic structures of right angled Artin groups by computing their series, Lie algebras, and cohomology, revealing relationships among growth series and algebraic invariants.

## Contribution

It provides explicit computations of the $p$-central and exponent-$p$ series, and describes the associated Lie algebras and their connections to cohomology and polynomial rings.

## Key findings

- Computed the $p$-central and exponent-$p$ series for all right angled Artin groups
- Described the associated Lie algebras and their relation to cohomology rings
- Established relationships between growth series of algebraic objects

## Abstract

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 a partially commuting polynomial ring and power series ring. We finally show how the growth series of these various objects are related to each other.