  Let \Sigma_g be a closed orientable surface let Diff_0(\Sigma_g; area) be the
identity component of the group of area-preserving diffeomorphisms of \Sigma_g.
In this work we present an extension of Gambaudo-Ghys construction to the case
of a closed hyperbolic surface \Sigma_g, i.e. we show that every non-trivial
homogeneous quasi-morphism on the braid group on n strings of \Sigma_g defines
a non-trivial 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 from Z^m to 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).
