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

Let w Fk be a nontrivial word and denote by w(G) G the image of the associated word map w: Gk G. Let G be one of the finite groups Sn,GLn(q),Sp2n(q),GO2n±(q),GO2n+1(q),GUn(q) (q a prime power, n 2, n 1), or the unitary group Un over . Let dG 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(G) for the degree of G.

For 𝜀 > 0, we prove that there exists an integer N(𝜀,w) such that w(G) is 𝜀-dense in G with respect to the metric dG if n(G) N(𝜀,w). 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(G) along the ultrafilter.

As a consequence of our methods, we also obtain an alternative proof of the result of Hui, Larsen, and Shalev that w1(SUn)w2(SUn) = SUn for nontrivial words w1,w2 Fk and n sufficiently large.

word map, word image, finite groups of Lie type, symmetric groups, Hamming metric, cohomology
Mathematical Subject Classification 2010
Primary: 20B30, 20D06, 20H30
Received: 13 November 2019
Revised: 2 April 2020
Accepted: 19 May 2020
Published: 31 July 2021
Jakob Schneider
Institut für Geometrie
TU Dresden
Andreas Thom
Institut für Geometrie
TU Dresden