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

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