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Word maps and Waring type problems

Author(s): Michael Larsen; Aner Shalev
Journal: J. Amer. Math. Soc.
MSC (2000): Primary 20D06, 20G40; Secondary 14G15
Posted: September 12, 2008
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Abstract: Waring's classical 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 nonabelian groups and simple groups in particular. Here the $ k$th power word is replaced by an arbitrary group word $ w \ne 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 various finite simple groups, showing that a product of length two suffices to express all elements. We also show that the set of values of $ w$ is very large, improving various results obtained so far.

Along the way we also obtain new results of independent interest on character values and class squares in symmetric groups.

Our methods involve algebraic geometry, representation theory, probabilistic arguments, as well as results from analytic number theory, including three primes theorems (approximating Goldbach's Conjecture).

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Additional Information:

Michael Larsen
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: larsen@math.indiana.edu

Aner Shalev
Affiliation: Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel
Email: shalev@math.huji.ac.il

DOI: 10.1090/S0894-0347-08-00615-2
PII: S 0894-0347(08)00615-2
Keywords: Word maps, finite simple groups, Waring's problem
Received by editor(s): February 1, 2007
Posted: September 12, 2008
Additional Notes: The first author was partially supported by NSF grant DMS-0354772
The second author was partially supported by an Israel Science Foundation Grant.
Both authors were partially supported by a Bi-National Science Foundation United States-Israel Grant.
Copyright of article: Copyright 2008, American Mathematical Society

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