## Geometry & Topology

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

#### 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 $Lp$–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.

#### Article information

Source
Geom. Topol., Volume 19, Number 6 (2015), 3289-3306.

Dates
Revised: 2 February 2015
Accepted: 6 April 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858875

Digital Object Identifier
doi:10.2140/gt.2015.19.3289

Mathematical Reviews number (MathSciNet)
MR3447104

Zentralblatt MATH identifier
1351.20021

#### Citation

Kim, Sang-hyun; Koberda, Thomas. Anti-trees and right-angled Artin subgroups of braid groups. Geom. Topol. 19 (2015), no. 6, 3289--3306. doi:10.2140/gt.2015.19.3289. https://projecteuclid.org/euclid.gt/1510858875

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