Word maps, conjugacy classes, and a noncommutative Waring-type theorem | Annals of Mathematics

Abstract Let w=w(x1,…,xd)≠1 be a nontrivial group word. We show that if G is a sufficiently large finite simple group, then every element g∈G can be expressed as a product of three values of w in G. This improves many known results for powers, commutators, as well as a theorem on general words obtained in [19]. The proof relies on probabilistic ideas, algebraic geometry, and character theory. Our methods, which apply the `zeta function’ ζG(s)=∑χ∈IrrGχ(1)−s, give rise to various additional results of independent interest, including applications to conjectures of Ore and Thompson.

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数学年刊(Annals of Mathematics)

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