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Medientyp:
E-Artikel
Titel:
Word maps, conjugacy classes, and a noncommutative Waring-type theorem
Beteiligte:
Shalev, Aner
Erschienen:
Dept. of Mathematics, Princeton University, 2009
Erschienen in:
Annals of Mathematics, 170 (2009) 3, Seite 1383-1416
Sprache:
Englisch
ISSN:
0003-486X
Entstehung:
Anmerkungen:
Beschreibung:
Let $w=w(x_{1},\ldots ,x_{d})\neq 1$ be a nontrivial group word. We show that if G is a sufficiently large finite simple group, then every element g ϵ 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' $\zeta _{G}(s)=\sum_{\chi \in \text{Irr}\ G}\chi (1)^{-s}$ , give rise to various additional results of independent interest, including applications to conjectures of Ore and Thompson.