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.

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有限单群的概率Waring问题

有限单群的概率 Waring 问题询问 $w_1w_2$ 形式的每个词（其中 $w_1$ 和 $w_2$ 是不相交的变量集中的非平凡词）是否在有限单群上相对于$L^1$ 标准。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。我们还提供了在有限秩的 Lie 类型的有限单群上引起几乎均匀分布的单词的几何特征，并研究了相关的随机游走。我们的第二个主要结果涉及有限单群的概率 $L^{\infty}$ Waring 问题。我们证明对于每个 $l \ge 1$ 都存在 $N = N(l)$，这样如果 $w_1, \ldots , w_N$ 是不相交的成对不相交集合中长度至多 $l$ 的非平凡词变量，那么他们的乘积 $w_1 \cdots w_N$ 在有限单群上就 $L^{\infty}$ 范数而言几乎是一致的。$N$ 对 $l$ 的依赖是真实的。这个结果意味着，对于上面的每个词 $w = w_1 \cdots w_N$，由 $w$ 在任意域上的半单代数群上引起的词映射是一个平面态射。还介绍了表示变体、子组增长和随机生成的应用。