The Waring problem for finite simple groups

Abstract

The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for non-abelian groups, and simple groups in particular. Here the k-th power word can be replaced by an arbitrary group word w = 1, and the goal is to express group elements as short products of values of w.We give a best possible and somewhat surprising solution for this Waring type problem for (non-abelian) finite simple groups of sufficiently high order, showing that a product of length two suffices to express all elements.Along the way we also obtain new results, possibly of independent interest, on character values in classical groups over finite fields, on regular semisimple elements lying in the image of word maps, and on products of conjugacy classes.Our methods involve algebraic geometry and representation theory, especially Lusztig's theory of representations of groups of Lie type.

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

“…The first, due to Nikolov and Pyber [98], is based on a recent result of Gowers [50], and the second, for finite simple groups of bounded Lie rank, due to Macpherson and Tent [88], relies on model theory. This result was substantially improved by Larsen, Shalev and Tiep in [70,71,73].…”

“…The words of the form = ∈ F 2 have also attracted special interest. Larsen, Shalev and Tiep [73] proved that any such word is surjective on sufficiently large finite simple groups. By further recent results of Guralnick and Malle [53] and of Liebeck, O'Brien, Shalev and Tiep [81], some words of the form are known to be surjective on all finite simple groups.…”

“…The most conclusive one, due to Larsen, Shalev and Tiep, says that for every nontrivial word there exists a constant C ( ) such that if G is a finite simple group satisfying # G > C ( ) then 2 (G) = G, see [71,73,74,114] and subsection 5.2.1 below.…”

“…Later on, Larsen, Shalev and Tiep [73] completed the proof for finite simple groups of Lie type of arbitrary rank.…”

Non-abelian Littlewood–Offord inequalities

“…(ii) As mentioned in the proof of [LST,Proposition 6.2.1], |C G (x)| ≤ q d 2 −2d+2 . As |χ(x)| ≤ |C G (x)| 1/2 , the claim follows for H = G. The claim for H = S follows in a similar fashion, by noting that…”