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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

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.

right-angled Artin group, braid group, cancellation theory, hyperbolic manifold, quasi-isometry
Mathematical Subject Classification 2010
Primary: 20F36
Secondary: 53D05, 20F10, 20F67
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
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