p -群的 wreath 积中的 Engel 概率
Engel probability in wreath products of $p$-groups
浏览论文内容
中文总结 AI 辅助
研究群\(W_k=(C_p\wr C_{p^k})^2\)中方程\(e_n(x,y)=g\)解的数量上下界，得出推论，证明了关于\(W_k\)中 Engel 字的更强猜想，表明其在特定有限群中不是概率恒等式，并构造相关闭子集。
中文摘要 AI 辅助
我们给出了群\(W_k=(C_p\wr C_{p^k})^2\)中方程\(e_n(x,y)=g\)的解的数量的上下界，其中\(e_n(x,y)\)是第\(n\)个 Engel 字且\(g\in W_k\)。由此得出几个推论。首先，我们证明了关于\(W_k\)中 Engel 字的 Amit - Ashurst 猜想的一个更强版本。还证明了在具有任意大 wreath 积商\(W_k\)的有限群中 Engel 字不是概率恒等式。最后，我们构造了\((C_p\wr\Z_p)^2\)的具有正 Haar 测度、空内部且是 Engel 字映射的原像的闭子集。
英文摘要
We give upper and lower bounds for the number of solutions of the equation $e_n(x,y) = g$ in the group $W_k=(C_p\wr C_{p^k})^2$, where $e_n(x,y)$ is the $n$-th Engel word and $g\in W_k$. We obtain several corollaries from this. First, we prove a stronger version of the Amit-Ashurst conjecture for Engel words in $W_k$. We also prove that Engel words are not probabilistic identities in profinite groups with arbitrarily large wreath product quotients $W_k$. To conclude, we construct closed subsets of $(C_p\wr\Z_p)^2$ with positive Haar measure, empty-interior, and which are the preimage of an Engel word map.
