Geometric embedding properties of Bestvina-Brady subgroups

Abstract

We compute the relative divergence 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. 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.

14 pages, To appear in Algebraic & Geometric Topology

References in corpus (1)

• Two-generator subgroups of right-angled Artin groups are quasi-isometrically embedded

Cited by in corpus (6)

• Coarse-median preserving automorphisms
• Malnormality and join-free subgroups in right-angled Coxeter groups
• Graphical splittings of Artin kernels
• Distortion of surfaces in graph manifolds
• Complete classification of the Dehn functions of Bestvina-Brady groups
• Quasi-isometry of pairs: surfaces in graph manifolds