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

Jakob Schneider and Andreas Thom

Vol. 311 (2021), No. 2, 475–504

DOI: 10.2140/pjm.2021.311.475

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 $S_n,{GL}_n\left(q\right),{Sp}_{2n^{\prime }}\left(q\right),{GO}_{2n^{\prime }}^{\pm }\left(q\right),{GO}_{2n^{\prime }+1}\left(q\right),{GU}_n\left(q\right)$ ($q$ 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

Milestones

Received: 13 November 2019

Revised: 2 April 2020

Accepted: 19 May 2020

Published: 31 July 2021