  In this work we construct Calabi quasi-morphisms on the universal cover of
the group Ham(M) of Hamiltonian diffeomorphisms for some non-monotone
symplectic manifolds. This complements a result by Entov and Polterovich which
applies in the monotone case. Moreover, in contrast to their work, we show that
these quasi-morphisms descend to non-trivial homomorphisms on the fundamental
group of Ham(M).
