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

Abstract

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 quasi-morphisms vanishing on all autonomous diffeomorphisms in the above group is infinite dimensional.