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

Abstract

On the group Symp(D, ∂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 quasimorphisms on the whole group Symp(D) of symplectomorphisms of the disk. As a corollary, we show that the second bounded cohomology H 2 b (Symp(D)) is infinitedimensional.