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

• 成果类型: Article
• 署名作者: Shalev, Aner
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.170.1383
• 发表日期: 2009
• 页码: 1383-1416
• 关键词: finite simple-groups; lie type; chevalley-groups; profinite groups; fuchsian-groups; random-walks; generation; PRODUCTS; GROWTH; POWERS

摘要

Let w = w (x(1), ..., x(d)) not equal 1 be a nontrivial group word. We show that if G is a sufficiently large finite simple group, then every element g is an element of 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) = Sigma(chi is an element of Irr G) chi(1)(-s), give rise to various additional results of independent interest, including applications to conjectures of Ore and Thompson.