Right-angled Artin groups (RAAGs) form a mysterious and rich class of groups that interpolates between the free groups and the free abelian groups. They are defined by a given graph, with generators corresponding to its vertices and commutator relations imposed for adjacent vertices. Unlike the case of free (abelian) groups, a subgroup of a RAAG may not be RAAG itself. In this direction, in 1987, Carl Droms characterised the graphs for which every subgroup of the associated RAAG is itself a RAAG.
In this talk we will deal with a similar problem for certain normal subgroups of RAAGs, called Bestvina-Brady groups, which have been extensively studied over the last 20 years, as they provide counterexamples to several group-theoretic problems.