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 , there is a free subgroup of rank 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 such that some right-angled Artin group has a free subgroup of rank whose inclusion is not a quasi-isometric embedding. It is well known that a right-angled Artin group is the fundamental group of a graph manifold whenever the defining graph is a tree with at least three vertices. We show that the Bestvina–Brady subgroup in this case is a horizontal surface subgroup.
"Geometric embedding properties of Bestvina–Brady subgroups." Algebr. Geom. Topol. 17 (4) 2499 - 2510, 2017. https://doi.org/10.2140/agt.2017.17.2499