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

TL;DR

This paper investigates the relationship between quasi-morphisms on braid groups and those on the group of area-preserving diffeomorphisms of the 2-disk, proving the injectivity of a key homomorphism connecting them.

Contribution

It establishes the injectivity of a homomorphism from quasi-morphisms on braid groups to those on the diffeomorphism group of the disk.

Findings

Proves the homomorphism is injective.

Connects braid group quasi-morphisms to diffeomorphism quasi-morphisms.

Enhances understanding of the algebraic structure of area-preserving diffeomorphisms.

Abstract

Recently Gambaudo and Ghys proved that there exist infinitely many quasi-morphisms on the group ${\rm Diff}_\Omega^\infty (D^2, \partial D^2)$ 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 ${\rm Diff}_\Omega^\infty (D^2, \partial D^2)$. In this paper, we study the homomorphism and prove its injectivity.