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.