Quantitative Representation Stability Over Linear Groups
Quantitative Representation Stability Over Linear Groups
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DOI：
10.1093/imrn/rny250
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发表时间：
2017-09
影响因子：
1
通讯作者：
Jeremy Miller;Jenny Wilson
中科院分区：
文献类型：
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作者：
Jeremy Miller;Jenny Wilson
我们引入了一种技术，用于证明特征零域上某些有限线性群的表示序列的定量表示稳定性定理。特别是，我们证明了 VIC 和 SI 模的更高 syzygies 的消失结果，这可以被认为是 FI 模背景下 Church-Ellenberg 正则定理的较弱版本。我们将这些技术应用于映射类群的同余子群和自由群的自同构群的同余子群的有理同源性。这部分解决了 Church 和 Putman-Sam 提出的问题。我们还证明了用扭曲系数映射自由群的类群和自同构群的新同调稳定性结果。
We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher syzygies of VIC- and SI-modules, which can be thought of as a weaker version of a regularity theorem of Church-Ellenberg in the context of FI-modules. We apply these techniques to the rational homology of congruence subgroups of mapping class groups and congruence subgroups of automorphism groups of free groups. This partially resolves a question raised by Church and Putman-Sam. We also prove new homological stability results for mapping class groups and automorphism groups of free groups with twisted coefficients.