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

Abstract

Let Σg be a closed orientable surface of genus g and let Diff 0(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 Diff 0(Σg, area ). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff 0(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).