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

Abstract

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