  These notes are the English version of the paper "Hyperbolicit\'e du graphe
des rayons et quasi-morphismes sur un gros groupe modulaire".
  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.
