#### Volume 19, issue 6 (2015)

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Anti-trees and right-angled Artin subgroups of braid groups

### Sang-hyun Kim and Thomas Koberda

Geometry & Topology 19 (2015) 3289–3306
##### Abstract

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree, and, consequently, into a pure braid group. It follows that $G$ is a quasi-isometrically embedded subgroup of the area-preserving diffeomorphism groups of the $2$–disk and of the $2$–sphere with ${L}^{p}$–metrics for suitable $p$. Another corollary is that there exists a closed hyperbolic manifold group of each dimension which admits a quasi-isometric group embedding into a pure braid group. Finally, we show that the isomorphism problem, conjugacy problem, and membership problem are unsolvable in the class of finitely presented subgroups of braid groups.

##### Keywords
right-angled Artin group, braid group, cancellation theory, hyperbolic manifold, quasi-isometry
##### Mathematical Subject Classification 2010
Primary: 20F36
Secondary: 53D05, 20F10, 20F67
##### Publication
Received: 27 May 2014
Revised: 2 February 2015
Accepted: 6 April 2015
Published: 6 January 2016
Proposed: Danny Calegari
Seconded: Leonid Polterovich, Martin R Bridson
##### Authors
 Sang-hyun Kim Department of Mathematical Sciences Seoul National University Seoul 151-747 South Korea Thomas Koberda Department of Mathematics University of Virginia Charlottesville, VA 22904-4137 USA