On universal right angled Artin groups

25 November 2014
Ashot Minasyan
A right angled Artin group (RAAG), also called a graph group or a partially commutative group, is a group which has a finite presentation where 
the only permitted defining relators are commutators of the generators. These groups and their subgroups play an important role in Geometric Group Theory, especially in view of the recent groundbreaking results of Haglund, Wise, Agol, and others, showing that many groups possess finite index subgroups that embed into RAAGs.
In their recent work on limit groups over right angled Artin groups, Casals-Ruiz and Kazachkov asked whether for every natural number n there exists a single "universal" RAAG, A_n, containing all n-generated subgroups of RAAGs. Motivated by this question, I will discuss several results showing that "universal" (in various contexts) RAAGs generally do not exist. I will also mention some positive results about universal groups for finitely presented n-generated subgroups of direct products of free and limit groups.