Geometric embedding properties of Bestvina-Brady subgroups

TL;DR

This paper investigates the geometric properties of Bestvina-Brady subgroups within right-angled Artin groups, focusing on divergence, distortion, and embedding characteristics, and provides new insights into subgroup structures and their geometric embeddings.

Contribution

It computes divergence and distortion of Bestvina-Brady subgroups and constructs free subgroups with non-quasi-isometric embeddings, answering a question about subgroup ranks in right-angled Artin groups.

Findings

Computed relative divergence and subgroup distortion of Bestvina-Brady subgroups.

Constructed free subgroups of rank n with non-quasi-isometric embeddings for n ≥ 3.

Identified Bestvina-Brady subgroups as horizontal surface subgroups in certain graph manifolds.

Abstract

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.