Quasi-morphisms and L^p-metrics on groups of volume-preserving diffeomorphisms

TL;DR

This paper investigates the relationship between quasi-morphisms and L^p-metrics on the group of volume-preserving diffeomorphisms of a manifold, establishing Lipschitz properties and constructing bi-Lipschitz embeddings under certain conditions.

Contribution

It proves that homogeneous quasi-morphisms induced by fundamental group quasi-morphisms are Lipschitz with respect to the L^p-metric, enabling bi-Lipschitz embeddings into the diffeomorphism group.

Findings

Homogeneous quasi-morphisms are Lipschitz with respect to the L^p-metric.

Bi-Lipschitz embeddings of finite-dimensional vector spaces are constructed.

Results depend on conditions related to the fundamental group.

Abstract

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. We show that every homogeneous quasi-morphism on the identity component $Diff_0(M,vol)$ of the group of volume preserving diffeomorphisms of M, which is induced by a quasi-morphism on the fundamental group, is Lipschitz with respect to the L^p-metric on the group $Diff_0(M,vol)$. As a consequence, assuming certain conditions on the fundamental group, we construct bi-Lipschitz embeddings of finite dimensional vector spaces into $Diff_0(M,vol)$.