Geometric Embedding Properties of Bestvina-brady Subgroups

We compute the relative divergence and the subgroup distortion of Bestvina-Brady subgroups. We also show that for each integer n ≥ 3, there is a free subgroup of rank n of some right-angled Artin group whose inclusion is not a quasi-isometric embedding. This result answers the question of Carr about the minimum rank n such that some rightangled Artin group has a free subgroup of rank n whose inclusion is not a quasi-isometric embedding. It is well-known that a right-angled Artin group AΓ is the fundamental group of a graph manifold whenever the defining graph Γ is a tree. We show that the Bestvina-Brady subgroup HΓ in this case is a horizontal surface subgroup.