#### Volume 17, issue 4 (2017)

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Geometric embedding properties of Bestvina–Brady subgroups

### Hung Cong Tran

Algebraic & Geometric Topology 17 (2017) 2499–2510
##### 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.

##### Keywords
Bestvina–Brady subgroups, geometric embedding properties, subgroup distortion, relative divergence
##### Mathematical Subject Classification 2010
Primary: 20F65, 20F67
Secondary: 20F36