Embedding problem between right-angled Artin groups

Abstract

Given a graph $\Gamma$, the right-angled Artin group $A(\Gamma )$ is given by the presentation $⟨u\in V(\Gamma )\mid [u,v]=1,\{u,v\}\in E(\Gamma )⟩$. The Embedding Problem in right-angled Artin groups asks, given two finite graphs $\Phi ,\Psi$, how to determine whether or not $A(\Phi )$ is isomorphic to a subgroup of $A(\Psi )$. 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.

20 pages, 11 figures. Notes of a mini-course given at the CIRM in April 2022 for the thematic semester "Groups Acting on Fractals, Hyperbolicity and Self-Similarity" of the Institut Henri Poincaré