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 $G:=Diff\left(D^2,area\right)$ be the group of smooth compactly supported area-preserving diffeomorphisms of $D^2$. For every natural number $k$ we construct an injective homomorphism $Z^k\to G$, 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 quasimorphisms vanishing on all autonomous diffeomorphisms in the above group is infinite-dimensional.