拟随机性与均匀孪生宽度
Quasirandomness and Uniform Twin-Width
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中文总结 AI 辅助
本文研究有限群的拟随机度与均匀孪生宽度的关系，证明前者给出后者的多项式下界、最小忠实复表示次数给出线性上界，非阿贝尔有限单群中二者多项式等价，还构造了特定有限表现群并确定三个Thompson群的均匀孪生宽度。
中文摘要 AI 辅助
对于每个非平凡有限群，我们证明其拟随机度给出了均匀孪生宽度的多项式下界，而其最小忠实复表示的次数给出了线性上界。对于非阿贝尔有限单群，这两个参数重合，因此在这类群中，均匀孪生宽度与拟随机性是多项式等价的，这在Gowers的意义下给出了拟随机性的一个新定义。我们利用该下界证明均匀孪生宽度在有限群上是无界的，这有助于我们构造具有有限孪生宽度但无限均匀孪生宽度的有限表现群，从而回答了Bonnet、Geniet、Tessera和Thomasse提出的问题。最后，我们确定了所有三个Thompson群的均匀孪生宽度。
英文摘要
For every nontrivial finite group, we prove that its quasirandom degree gives a polynomial lower bound on its uniform twin-width, whereas its minimum faithful complex representation degree gives a linear upper bound. For nonabelian finite simple groups, these two parameters coincide, so uniform twin-width is polynomially equivalent to quasirandomness in that class, yielding a new definition of quasirandomness in the sense of Gowers. We use the lower bound to prove that uniform twin-width is unbounded over finite groups, which helps us construct finitely presented groups with finite twin-width but infinite uniform twin-width. This answers a question of Bonnet, Geniet, Tessera and Thomasse. Finally, we determine the uniform twin-width of all three Thompson groups.
