Word maps, conjugacy classes, and a noncommutative waring-type theorem

Aner Shalev*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

75 Scopus citations


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)= Σχ∈Irr G χ(1)-s, give rise to various additional results of independent interest, including applications to conjectures of Ore and Thompson.

Original languageAmerican English
Pages (from-to)1383-1416
Number of pages34
JournalAnnals of Mathematics
Issue number3
StatePublished - 2009


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