Quasi-isometric embeddings into diffeomorphism groups

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group π1(M) we construct quasi-isometric embeddings of either free Abelian or direct products of non-Abelian free groups into the group of volume preserving diffeomorphisms of M equipped with the Lp metric induced by a Riemannian metric on M.