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.
Bibliographical noteBoth authors would like to thank the anonymous referee for careful reading of our paper and for his/her helpful comments and remarks. This work has been done during the first author stay in Aberdeen and in Mathematisches Forschungsinstitut Oberwolfach. The first author wishes to express his gratitude to both institutes. He was supported by ESF grant number 4824 and by the Oberwolfach Leibniz fellowship. The visit in Aberdeen was supported by the CAST network.
- area-preserving diffeomorphisms
- braid groups
- quasi-isometric embeddings
- bi-invariant metrics