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Let $w = w(x_1,\ldots , x_d) \ne 1$ be a nontrivial group word. We show that if $G$ is a sufficiently large finite simple group, then every element $g \in G$ 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’ $\zeta_G(s) = \sum_{\chi \in {\rm Irr}\, G} \chi(1)^{-s}$, give rise to various additional results of independent interest, including applications to conjectures of Ore and Thompson.
@article{10_4007_annals_2009_170_1383,
author = {Aner Shalev},
title = {Word maps, conjugacy classes, and a noncommutative {Waring-type} theorem},
journal = {Annals of mathematics},
pages = {1383--1416},
publisher = {mathdoc},
volume = {170},
number = {3},
year = {2009},
doi = {10.4007/annals.2009.170.1383},
mrnumber = {2600876},
zbl = {1203.20013},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.170.1383/}
}
TY - JOUR AU - Aner Shalev TI - Word maps, conjugacy classes, and a noncommutative Waring-type theorem JO - Annals of mathematics PY - 2009 SP - 1383 EP - 1416 VL - 170 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.170.1383/ DO - 10.4007/annals.2009.170.1383 LA - en ID - 10_4007_annals_2009_170_1383 ER -
%0 Journal Article %A Aner Shalev %T Word maps, conjugacy classes, and a noncommutative Waring-type theorem %J Annals of mathematics %D 2009 %P 1383-1416 %V 170 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.170.1383/ %R 10.4007/annals.2009.170.1383 %G en %F 10_4007_annals_2009_170_1383
Aner Shalev. Word maps, conjugacy classes, and a noncommutative Waring-type theorem. Annals of mathematics, Tome 170 (2009) no. 3, pp. 1383-1416. doi: 10.4007/annals.2009.170.1383
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