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

Let w D w.x 1 ; : : : ; x d / 1 be a nontrivial group word. We show that if G is a sufficiently large finite simple group, then every element g 2 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 The proof relies on probabilistic ideas, algebraic geometry, and character theory. Our methods, which apply the 'zeta function' G .s/ D P 2Irr G .1/ s , give rise to various additional results of independent interest, including applications to conjectures of Ore and Thompson. 1. Main result 2. Intermediate results 3. Alternating groups 4. Character theoretic preparations 5. Almost uniform distributions, I 6. Proof of main result 7. Class expansion 8. Almost uniform distributions, II 9. Conjectures of Ore and Thompson 10. Open problems and examples References