Probabilistic Waring problems for finite simple groups
Probabilistic Waring problems for finite simple groups
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DOI：
10.4007/annals.2019.190.2.3
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发表时间：
2018-08
影响因子：
4.9
通讯作者：
M. Larsen;A. Shalev;P. Tiep
中科院分区：
文献类型：
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作者：
M. Larsen;A. Shalev;P. Tiep
有限单群的概率Wering问题是指，形式为$w1w2$的每一个词，其中$w1$和$w2$都是不相交变量集中的非平凡词，是否关于$L^1$范数在有限单群上诱导出几乎一致的分布.我们的第一个主要结果为这个问题提供了一个积极的解决方案。给出了有限有界秩李型单群上诱导几乎均匀分布的词的几何刻画，并研究了相关的随机游动。我们的第二个主要结果是关于有限单群的概率L^1问题。证明了对于每一个$L 1$，都存在$N=N(L)$，使得如果$w_1，点，w_N$是两两不相交变量集中长度至多为$L$的非平凡字，则它们的乘积$w_1\cdots w_N$在有限单群上关于$L^^#$范数几乎一致.$N$对$L$的依赖是真实的。这个结果表明，对于如上所述的词$w=w1\cdots w_N$，由$w$在任意域上的半单代数群上诱导的词映射是平坦的态射。文中还介绍了它在表示簇、子群增长和随机生成等方面的应用。
The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, where $w_1$ and $w_2$ are non-trivial words in disjoint sets of variables, induces almost uniform distribution on finite simple groups with respect to the $L^1$ norm. Our first main result provides a positive solution to this problem. We also provide a geometric characterization of words inducing almost uniform distribution on finite simple groups of Lie type of bounded rank, and study related random walks. Our second main result concerns the probabilistic $L^{\infty}$ Waring problem for finite simple groups. We show that for every $l \ge 1$ there exists $N = N(l)$, such that if $w_1, \ldots , w_N$ are non-trivial words of length at most $l$ in pairwise disjoint sets of variables, then their product $w_1 \cdots w_N$ is almost uniform on finite simple groups with respect to the $L^{\infty}$ norm. The dependence of $N$ on $l$ is genuine. This result implies that, for every word $w = w_1 \cdots w_N$ as above, the word map induced by $w$ on a semisimple algebraic group over an arbitrary field is a flat morphism. Applications to representation varieties, subgroup growth, and random generation are also presented.