  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.
