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

Kim, Sang-hyun¹ ; Koberda, Thomas²

¹ Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea
² Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA

Geometry & topology, Tome 19 (2015) no. 6, pp. 3289-3306.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.3289

Classification : 20F36, 53D05, 20F10, 20F67
Keywords: right-angled Artin group, braid group, cancellation theory, hyperbolic manifold, quasi-isometry

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