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

Let w = w(x₁,...,x_d) ≠ 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)= Σ_{χ∈Irr G} χ(1)^-s, give rise to various additional results of independent interest, including applications to conjectures of Ore and Thompson.