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

Hyperbolicit\'e du graphe des rayons et quasi-morphismes sur un gros groupe modulaire