Subgroups of Bestvina-Brady groups

In 1987, Droms [15] proved that all the subgroups of a right-angled Artin group (RAAG) defined by a finite simple graph Γ are themselves RAAGs if, and only if, Γ has no induced squares nor lines of length 3. The present work provides a similar result for specific normal subgroups of RAAGs, called Bestvina-Brady groups: We characterize those finite graphs in which every subgroup of such a group is itself a RAAG. In turn, we confirm several Galois theoretic conjectures for the pro- p analogues of these groups, and study their associated graded Lie algebras.