#### Vol. 311, No. 2, 2021

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

### Jakob Schneider and Andreas Thom

Vol. 311 (2021), No. 2, 475–504
##### Abstract

Let $w\in {F}_{k}$ be a nontrivial word and denote by $w\left(G\right)\subseteq G$ the image of the associated word map $w:{G}^{k}\to G$. Let $G$ be one of the finite groups ${S}_{n},{GL}_{n}\left(q\right),{Sp}_{2{n}^{\prime }}\left(q\right),{GO}_{2{n}^{\prime }}^{±}\left(q\right),{GO}_{2{n}^{\prime }+1}\left(q\right),{GU}_{n}\left(q\right)$ ($q$ a prime power, $n\ge 2$, ${n}^{\prime }\ge 1$), or the unitary group ${U}_{n}$ over $ℂ$. Let ${d}_{G}$ be the normalized Hamming metric resp. the normalized rank metric on $G$ when $G$ is a symmetric group resp. one of the other classical groups and write $n\left(G\right)$ for the degree of $G$.

For $𝜀>0$, we prove that there exists an integer $N\left(𝜀,w\right)$ such that $w\left(G\right)$ is $𝜀$-dense in $G$ with respect to the metric ${d}_{G}$ if $n\left(G\right)\ge N\left(𝜀,w\right)$. This confirms metric versions of conjectures by Shalev and Larsen. Equivalently, we prove that any nontrivial word map is surjective on a metric ultraproduct of groups $G$ from above such that $n\left(G\right)\to \infty$ along the ultrafilter.

As a consequence of our methods, we also obtain an alternative proof of the result of Hui, Larsen, and Shalev that ${w}_{1}\left({SU}_{n}\right){w}_{2}\left({SU}_{n}\right)={SU}_{n}$ for nontrivial words ${w}_{1},{w}_{2}\in {F}_{k}$ and $n$ sufficiently large.

##### Keywords
word map, word image, finite groups of Lie type, symmetric groups, Hamming metric, cohomology
##### Mathematical Subject Classification 2010
Primary: 20B30, 20D06, 20H30