Extensions of quasi-morphisms to the symplectomorphism group of the disk

TL;DR

This paper demonstrates that certain quasi-morphisms on the symplectomorphism group of the disk extend from the boundary-identity subgroup to the entire group, revealing the infinite-dimensionality of its second bounded cohomology.

Contribution

It proves the extension of Ruelle and Gambaudo-Ghys quasi-morphisms to the full symplectomorphism group of the disk, a novel result in symplectic topology.

Findings

Quasi-morphisms extend to the whole symplectomorphism group.

Second bounded cohomology of the group is infinite-dimensional.

Implications for symplectic topology and group cohomology.

Abstract

On the group $\rm{Symp}(D, \partial D)$ of symplectomorphisms of the disk which are the identity near the boundary, there are homogeneous quasi-morphisms called the Ruelle invariant and Gambaudo-Ghys quasi-morphisms. In this paper, we show that the above homogeneous quasi-morphisms extend to homogeneous quasi-morphisms on the whole group $\rm{Symp}(D)$ of symplectomorphisms of the disk. As a corollary, we show that the second bounded cohomology $H_b^2(\rm{Symp}(D))$ is infinite-dimensional.