Commutators in mapping class groups and bounded cohomology
Commutators in mapping class groups and bounded cohomology
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映射类群和有界上同调中的交换子
DOI：
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发表时间：
2000
期刊：
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通讯作者：
Mustafa Korkmaz
中科院分区：
文献类型：
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作者：
Mustafa Korkmaz
我们证明了Dehn twist的稳定换位子长度是正的，这意味着从第二有界上同调到普通上同调的自然映射不是单射的，映射类群不是一致完美的，Dehn twist的增长率是线性的.然后证明了超椭圆映射类群的第二有界上同调维数和亏格至多为2的曲面的映射类群的第二有界上同调维数等于连续统的基数.对任意正整数n，给出亏格为2的曲面的映射类群的子群Gn，使得Gn不含Torelli群且其第一上同调的秩至少为n.
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.