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

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. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk and the 2--sphere, answering questions due to Crisp--Wiest and M. Kapovich. Another corollary is that a pure braid group contains a closed hyperbolic manifold group as a quasi-isometrically embedded subgroup up to dimension eight. Finally, we show that the isomorphism problem, conjugacy problem, and membership problems are unsolvable in the class of finitely presented subgroups of braid groups.