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Geometric embedding properties of Bestvina–Brady subgroups

Hung Cong Tran

Algebraic & Geometric Topology 17 (2017) 2499–2510

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 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 AΓ is the fundamental group of a graph manifold whenever the defining graph Γ is a tree with at least three vertices. We show that the Bestvina–Brady subgroup HΓ in this case is a horizontal surface subgroup.

Bestvina–Brady subgroups, geometric embedding properties, subgroup distortion, relative divergence
Mathematical Subject Classification 2010
Primary: 20F65, 20F67
Secondary: 20F36
Received: 23 August 2016
Revised: 20 October 2016
Accepted: 1 January 2017
Published: 3 August 2017
Hung Cong Tran
Department of Mathematics
The University of Georgia
Athens, GA
United States