On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc

TL;DR

This paper studies the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc, revealing its rich algebraic structure and the infinite-dimensional space of related quasi-morphisms.

Contribution

It constructs bi-Lipschitz homomorphisms from free abelian groups into the diffeomorphism group and shows the space of certain quasi-morphisms is infinite dimensional.

Findings

Existence of bi-Lipschitz homomorphisms from Z^k to the diffeomorphism group.

The space of homogeneous quasi-morphisms vanishing on autonomous diffeomorphisms is infinite dimensional.

The autonomous metric induces a rich algebraic and geometric structure on the group.

Abstract

Let $D^2$ be the open unit disc in the Euclidean plane and let $G:= Diff(D2; area)$ be the group of smooth compactly supported area-preserving diffeomorphisms of $D^2$. We investigate the properties of G endowed with the autonomous metric. In particular, we construct a bi-Lipschitz homomorphism $Z^k \rightarrow G$ of a finitely generated free abelian group of an arbitrary rank. We also show that the space of homogeneous quasi-morphisms vanishing on all autonomous diffeomorphisms in the above group is infinite dimensional.