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• W2898189479 abstract "Let $winmathbf F_k$ be a non-trivial word and denote by $w(G)subseteq G$ the image of the associated word map $wcolon G^kto G$. 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, $ngeq 2$, $mgeq 1$), 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 $n(G)toinfty$ 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 $w_1,w_2inmathbf F_k$ and $n$ sufficiently large." @default.
• W2898189479 created "2018-11-02" @default.
• W2898189479 creator A5010043767 @default.
• W2898189479 creator A5082162969 @default.
• W2898189479 date "2021-07-31" @default.
• W2898189479 modified "2023-09-27" @default.
• W2898189479 title "Word images in symmetric and classical groups of Lie type are dense" @default.
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• W2898189479 doi "https://doi.org/10.2140/pjm.2021.311.475" @default.
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