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

Abstract

Let $D^2$ be the open unit disc in the Euclidean plane and let $G := Diff(D^2; area)$ be the group of smooth compactly supported area-preserving diffeomorphisms of $D^2$. For every natural number $k$ we construct an injective homomorphism $Z^k → G$, which is bi-Lipschitz with respect to the word metric on $Z^k$ and the autonomous metric on $G$. We also show that the space of homogeneous quasi-morphisms vanishing on all autonomous diffeomorphisms in the above group is infinite dimensional.