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 $n\geq 3$, 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 right-angled 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 $A_\Gamma$ is the fundamental group of a graph manifold whenever the defining graph $\Gamma$ is a tree. We show that the Bestvina-Brady subgroup $H_\Gamma$ in this case is a horizontal surface subgroup.