Geometry & Topology

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

Sang-hyun Kim and Thomas Koberda

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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

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

Received: 27 May 2014
Revised: 2 February 2015
Accepted: 6 April 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F36: Braid groups; Artin groups
Secondary: 53D05: Symplectic manifolds, general 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] 20F67: Hyperbolic groups and nonpositively curved groups

right-angled Artin group braid group cancellation theory hyperbolic manifold quasi-isometry


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.

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