Embedding problem between right-angled Artin groups

Abstract

Given a graph $Γ$, the right-angled Artin group $A(Γ)$ is given by the presentation $\langle u \in V(Γ) \mid [u,v]=1, \ \{u,v\} \in E(Γ) \rangle$. The Embedding Problem in right-angled Artin groups asks, given two finite graphs $Φ,Ψ$, how to determine whether or not $A(Φ)$ is isomorphic to a subgroup of $A(Ψ)$. These are the notes of a mini-course given at the CIRM in April 2022, dedicated to a geometric point of view on this embedding problem, based on quasi-median graphs.