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

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