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

Abstract

Let w ∈ F k be a non-trivial word and denote by w(G) ⊆ G the image of the associated word map w : G k → G. Let G be one of the finite groups Sn, GLn(q), Sp 2m (q), GO ± 2m (q), GO2m+1(q), GUn(q) (q a prime power, n ≥ 2, m ≥ 1), or the unitary group Un over C. Let dG 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 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 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) → ∞ along the ultrafilter.As a consequence of our methods, we also obtain an alternative proof of the result of Hui-Larsen-Shalev that w1(SUn)w2(SUn) = SUn for non-trivial words w1, w2 ∈ F k and n sufficiently large.