多项式置换稳定性、软性与通用多项式群
Polynomial permutation stability, soficity, and universal polynomial groups
AI总结：
该研究将置换稳定性扩展到群的多项式映射，引入通用多项式群$\text{Pol}_s(G)$，发展多项式软逼近理论，刻画有限群高次通用多项式群结构，得到可和性阻碍与特定有限群的 largeness 结果。
AI中文摘要：
我们引入多项式置换稳定性，将经典的置换稳定性从群同态扩展到群的多项式映射。通用多项式群$\boldsymbol{\text{Pol}}_s(G)$为该理论提供了自然框架，我们证明$s$次多项式稳定性等价于$\text{Pol}_s(G)$的置换稳定性，由此特别得到所有有限群在二次时的多项式稳定性。我们发展了对应的多项式软逼近理论，证明对于可数群，该理论未给出软性的新概念；但相关通用多项式群包含新的结构信息：二次时它们可嵌入一个 wreath 积，这使我们能根据原群刻画其软性。我们还通过通用多项式群的弱稳定性理论，公式化并研究弱多项式稳定性。对于有限群，我们探究高次通用多项式群的结构，得到可和性的一般阻碍，并对一类具有完全导出子群且阿贝尔化阶为2的有限群，证明其 largeness 结果。
英文摘要：
We introduce polynomial permutation stability, extending classical permutation stability from homomorphisms to polynomial maps of groups. The universal polynomial group $\operatorname{Pol}_s(G)$ provides a natural framework for this theory, and we prove that polynomial stability of degree $s$ is equivalent to permutation stability of $\operatorname{Pol}_s(G)$. This yields, in particular, polynomial stability of all finite groups in degree two. We develop a corresponding theory of polynomial sofic approximations and show that, for countable groups, it does not give a new notion of soficity. Nevertheless, the associated universal polynomial groups contain new structural information: in degree two they embed into a wreath product, which allows us to characterize their soficity in terms of the original group. We also formulate and study weak polynomial stability through the weak stability theory of universal polynomial groups. For finite groups, we investigate the structure of higher-degree universal polynomial groups. We obtain general obstructions to amenability and prove largeness results for a family of finite groups with perfect derived subgroup and abelianization of order two.
