### abstract

- Let D2 be the open unit disc in the Euclidean plane and let G WD 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 … Let D 2 R2 be the open unit disc and let HW D 2 ! R be a smooth compactly … XH .x;y/ D @H @y @x C @H @x @y … It is a well known fact that every smooth compactly supported and area-preserving diffeomorphism of the disc D 2 is a composition of finitely many autonomous dif- feomorphisms; see Banyaga [2]. How many? In the present paper we are interested in the geometry of this question. More precisely, we define the autonomous norm on the …