Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group $\textsl{Diff}(D^2,\partial D^2,\textsl{vol})$ of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of $F_n$ and $\mathbb{Z}^n$ for all $n>0$. As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the group $\textsl{Diff}(D^2,\partial D^2,\textsl{vol})$. Examples include hyperbolic surface groups, some HNN-extensions of these along cyclic subgroups and the fundamental group of a certain closed hyperbolic 3-manifold.