Quasi-isometric embeddings into diffeomorphism groups

Abstract

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 ${\pi }_1(M)$ 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 $L^p$ metric induced by a Riemannian metric on $M$.