TY - JOUR

T1 - Word maps, conjugacy classes, and a noncommutative waring-type theorem

AU - Shalev, Aner

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=71749090468&partnerID=8YFLogxK

U2 - 10.4007/annals.2009.170.1383

DO - 10.4007/annals.2009.170.1383

M3 - Article

AN - SCOPUS:71749090468

SN - 0003-486X

VL - 170

SP - 1383

EP - 1416

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 3

ER -