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 $L^p$–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.