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arXiv 1802.09289math.GR
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Lie型对称群和典型群中的词象是稠密的

Word images in symmetric and classical groups of Lie type are dense

Jakob Schneider, Andreas Thom

备注：
29 pages, no figures
发表信息：
Pacific J. Math. 311 (2021) 475-504
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摘要：

令$w\in\mathbf F_k$是一个非平凡词，并用$w(G)\subseteq G$表示关联词映射$w\colon G^k\to G$的图像。设$G$是有限群${\rm S}_n,{\rm GL}_n(q),{\rm Sp}_{2m}(q),{\rm GO}_{2m}^\pm(q),{\rm GO}_{2m+1}(q),{\rm GU}_n(q)$（$q$是素数幂，$n\geq 2$，$m\geq 1$）或$\mathbb C$上的酉群${\rm U}_n$之一。令$d_G$为归一化汉明距离，当$G$是对称群时，$G$上的归一化秩度量。其他典型群中的一个，并分别为置换写$n(G)$。$G$的Lie秩。对于$\varepsilon>0$，我们证明了存在一个整数$N(\varepsilon,w)$使得$w(G)$在$G$中关于度量$d_G$是$\varepsilon$ -稠密的，如果$n(G)\geq N(\varepsilon,w)$ .这证实了Shalev和Larsen的度量版本。等价地，我们证明了任何非平凡的词映射是满射的度量超积群$G$从上面这样的$n(G)\to\infty$沿超滤子。作为我们的方法的结果，我们也得到了另一种证明的结果Hui-Larsen-Shalev，$w_1({\rm SU}_n)w_2({\rm SU}_n)={\rm SU}_n$为非平凡的话$w_1,w_2\in\mathbf F_k$和$n$足够大。

英文摘要：

Let $w\in\mathbf F_k$ be a non-trivial word and denote by $w(G)\subseteq G$ the image of the associated word map $w\colon G^k\to G$. Let $G$ be one of the finite groups ${\rm S}_n,{\rm GL}_n(q),{\rm Sp}_{2m}(q),{\rm GO}_{2m}^\pm(q),{\rm GO}_{2m+1}(q),{\rm GU}_n(q)$ ($q$ a prime power, $n\geq 2$, $m\geq 1$), or the unitary group ${\rm U}_n$ over $\mathbb C$. Let $d_G$ be the normalized Hamming distance resp. the normalized rank metric on $G$ when $G$ is a symmetric group resp. one of the other classical groups and write $n(G)$ for the permutation resp. Lie rank of $G$. For $\varepsilon>0$, we prove that there exists an integer $N(\varepsilon,w)$ such that $w(G)$ is $\varepsilon$-dense in $G$ with respect to the metric $d_G$ if $n(G)\geq N(\varepsilon,w)$. This confirms metric versions of a conjectures by Shalev and Larsen. Equivalently, we prove that any non-trivial word map is surjective on a metric ultraproduct of groups $G$ from above such that $n(G)\to\infty$ along the ultrafilter. As a consequence of our methods, we also obtain an alternative proof of the result of Hui-Larsen-Shalev that $w_1({\rm SU}_n)w_2({\rm SU}_n)={\rm SU}_n$ for non-trivial words $w_1,w_2\in\mathbf F_k$ and $n$ sufficiently large.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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