Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces

Let Sigma(g) be a closed orientable surface of genus g and let Diff(0)(Sigma(g), area) be the identity component of the group of area-preserving diffeomorphisms of Sigma(g). In this paper, we present the extension of Gambaudo-Ghys construction to the case of a closed hyperbolic surface Sigma(g), i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of Sigma(g) defines a nontrivial homogeneous quasi-morphism on the group Diff(0)(Sigma(g), area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff(0)(Sigma(g), area) is infinite-dimensional. Let Ham(Sigma(g)) be the group of Hamiltonian diffeomorphisms of Sigma(g). As an application of the above construction we construct two injective homomorphisms Z(m) -> Ham(Sigma(g)), which are bi-Lipschitz with respect to the word metric on Z(m) and the autonomous and fragmentation metrics on Ham(Sigma(g)). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham(Sigma(g)).