Abstract. The mapping class group Γ 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 Γ on this graph to find an explicit non trivial quasimorphism on Γ and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a hyperbolic element of Γ with vanishing stable commutator length. This carries out a program proposed by Danny Calegari.
We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these 'big' mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.
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