Geometric embedding properties of Bestvina–Brady subgroups

Abstract

We compute the relative divergence of right-angled Artin groups with respect to their Bestvina–Brady subgroups and the subgroup distortion of Bestvina–Brady subgroups. We also show that for each integer $n\ge 3$, there is a free subgroup of rank $n$ of some right-angled Artin group whose inclusion is not a quasi-isometric embedding. The corollary 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 with at least three vertices. We show that the Bestvina–Brady subgroup $H_{\Gamma }$ in this case is a horizontal surface subgroup.