Quasi-Isometric Embeddings into Diffeomorphism Groups

TL;DR

This paper constructs quasi-isometric embeddings of free Abelian and free group products into the volume-preserving diffeomorphism groups of certain manifolds, revealing geometric properties of these groups.

Contribution

It introduces a method to embed free Abelian and non-Abelian free groups quasi-isometrically into diffeomorphism groups under specific conditions on the manifold's fundamental group.

Findings

Embedding of free Abelian groups into diffeomorphism groups.

Embedding of free non-Abelian groups into diffeomorphism groups.

Reveals geometric structure of diffeomorphism groups with L^p metrics.

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.