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

TL;DR

This paper proves that for large enough classical and symmetric groups, the images of non-trivial word maps are dense with respect to natural metrics, confirming conjectures and providing new proofs of existing results.

Contribution

It establishes metric density of word map images in large classical groups, confirming conjectures by Shalev and Larsen, and offers an alternative proof of a product surjectivity result for special unitary groups.

Findings

Word images are dense in groups with large rank.

Non-trivial word maps are surjective on ultraproducts of groups as rank tends to infinity.

Provides an alternative proof for product surjectivity in special unitary groups.

Abstract

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.