Let σg be a closed orientable surface of genus g and let Diff0(σg, area) be the identity component of the group of area-preserving diffeomorphisms of σg. In this paper, we present the extension of Gambaudo-Ghys construction to the case of a closed hyperbolic surface σg, i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of σg defines a nontrivial homogeneous quasi-morphism on the group Diff0 (σg, area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff0(σg, area) is infinite-dimensional. Let Ham(σg) be the group of Hamiltonian diffeomorphisms of σg. As an application of the above construction we construct two injective homomorphisms Zm → Ham(σg), which are bi-Lipschitz with respect to the word metric on Zm and the autonomous and fragmentation metrics on Ham(σg). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham(σg).
- Groups of Hamiltonian diffeomorphisms
- bi-invariant metrics
- braid groups
- mapping class groups