Hyperbolicity ray graph and quasi-morphisms on a large modular group

The mapping class group Gamma of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of Gamma on this graph to find an explicit non trivial quasimorphism on Gamma and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a hyperbolic element of Gamma with vanishing stable commutator length. This carries out a program proposed by Danny Calegari.