Subgroups of Bestvina-Brady groups

TL;DR

This paper characterizes graphs whose associated Bestvina-Brady groups have all subgroups as RAAGs and confirms related Galois theoretic conjectures for their pro-p completions.

Contribution

It extends subgroup classification results from RAAGs to specific normal subgroups called Bestvina-Brady groups and verifies Galois conjectures for their pro-p completions.

Findings

Characterization of graphs with all subgroups of Bestvina-Brady groups as RAAGs

Confirmation of Galois theoretic conjectures for pro-p completions of these groups

Extension of subgroup structure results from RAAGs to Bestvina-Brady groups

Abstract

In "Subgroups of Graph Groups", 1987, J. Alg., Droms proved that all the subgroups of a right-angled Artin group (RAAG) defined by a finite simplicial graph $\Gamma$ are themselves RAAGs if, and only if, $\Gamma$ has no induced square graph nor line-graph of length $3$. The present work provides a similar result for specific normal subgroups of RAAGs, called Bestvina-Brady groups: We characterize those graphs in which every subgroup of such a group is itself a RAAG. In turn, we confirm several Galois theoretic conjectures for the pro-$p$ completions of these groups.