Quasi-morphisms on the group of area-preserving diffeomorphisms of the $2$-disk via braid groups

Abstract:

Recently Gambaudo and Ghys proved that there exist infinitely many quasi-morphisms on the group $\textrm {Diff}_\Omega ^\infty (D^2, \partial D^2)$ of area-preserving diffeomorphisms of the $2$-disk $D^2$. For the proof, they constructed a homomorphism from the space of quasi-morphisms on the braid group to the space of quasi-morphisms on $\textrm {Diff}_\Omega ^\infty (D^2, \partial D^2)$. In this paper, we study this homomorphism and prove its injectivity.