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

TL;DR

This paper extends the Gambaudo-Ghys construction to hyperbolic surfaces, demonstrating the infinite-dimensionality of quasi-morphism spaces on surface diffeomorphism groups and constructing new bi-Lipschitz embeddings and Calabi quasi-morphisms.

Contribution

It generalizes quasi-morphism constructions to hyperbolic surfaces and introduces new embeddings and Calabi quasi-morphisms on Hamiltonian diffeomorphism groups.

Findings

Infinite-dimensional space of homogeneous quasi-morphisms on Diff_0( ext{surface})

Bi-Lipschitz embeddings of Z^m into Hamiltonian diffeomorphisms

New family of Calabi quasi-morphisms

Abstract

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).