线性代数群和置换群的增长性：走向统一的观点
Growth in linear algebraic groups and permutation groups: towards a unified perspective
摘要：
到目前为止，我们已经在每一个有限的李型单群$G$中得到了一个乘积定理，其界的强度只取决于$G$的秩。这样的定理有许多后果：凯莱图的直径的界限，谱间隙，等等。对于交错群Alt_n，我们有一个拟多对数直径界（Helfgott-Seress 2014），但它不依赖于乘积定理。我们将重新讨论Alt_n的界的证明，使它更接近线性代数群的证明，并使一些共同的主题更清楚。因此，我们将展示如何证明Alt_n的乘积定理-不是完全的强度，因为这是不可能的，但足够强以暗示直径界。
英文摘要：
By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Alt_n, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. We shall revisit the proof of the bound for Alt_n, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Alt_n -- not of full strength, as that would be impossible, but strong enough to imply the diameter bound.
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