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Tori Akin : Automorphisms of the Punctured Mapping Class Group

We can describe the point-pushing subgroup of the mapping class group topologically as the set of maps that push a puncture around loops in the surface. However, we can characterize this topological subgroup in purely algebraic terms. Using group theoretic tools and a classical theorem of Burnside, we can recover a result of Ivanov-McCarthy establishing the triviality of Out(Mod±). To this end, we’ll demonstrate that the point-pushing subgroup is “unique” in the mapping class group.

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