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

Let D2 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 D2. For every natural number k we construct an injective homomorphism Zk ¿ G, which is bi-Lipschitz with respect to the word metric on Zk 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.