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


Let $w = w(x_1,\ldots , x_d) \ne 1$ be a nontrivial group word. We show that if $G$ is a sufficiently large finite simple group, then every element $g \in G$ 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’ $\zeta_G(s) = \sum_{\chi \in {\rm Irr}\, G} \chi(1)^{-s}$, give rise to various additional results of independent interest, including applications to conjectures of Ore and Thompson.