Commutators in Mapping Class Groups and Bounded Cohomology
نویسنده
چکیده
We prove that the stable commutator lengths of Dehn twists are positive, implying that the natural map from the second bounded cohomology to the ordinary cohomology is not injective, that mapping class groups are not uniformly perfect and that the growth rate of Dehn twists are linear. We then prove that the dimensions of the second bounded cohomology of hyperelliptic mapping class groups and of the mapping class groups of surfaces of genus at most two are equal to the cardinal of the continuum. For any positive integer n, we exhibit a subgroup Gn of the mapping class group of a surface of genus two such that Gn does not contain the Torelli group and the rank of its first cohomology is at least n.
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