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 Diff Ω ∞ ( D 2 , ∂ D 2 ) \textrm {Diff}_\Omega ^\infty (D^2, \partial D^2) of area-preserving diffeomorphisms of the 2 2 -disk D 2 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 Diff Ω ∞ ( D 2 , ∂ D 2 ) \textrm {Diff}_\Omega ^\infty (D^2, \partial D^2) . In this paper, we study this homomorphism and prove its injectivity.