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Sieve methods in group theory I: Powers in linear groups


Authors: Alexander Lubotzky and Chen Meiri
Journal: J. Amer. Math. Soc. 25 (2012), 1119-1148
MSC (2010): Primary 20Pxx
DOI: https://doi.org/10.1090/S0894-0347-2012-00736-X
Published electronically: April 11, 2012
MathSciNet review: 2947947
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Abstract: A general sieve method for groups is formulated. It enables one to ``measure'' subsets of a finitely generated group. As an application we show that if $ \Gamma $ is a finitely generated non-virtually solvable linear group of characteristic zero, then the set of proper powers in $ \Gamma $ is exponentially small. This is a far-reaching generalization of a result of Hrushovski, Kropholler, Lubotzky, and Shalev.


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Additional Information

Alexander Lubotzky
Affiliation: Einstein Institute of Mathematics, Hebrew University, Jerusalem 90914, Israel
Email: alexlub@math.huji.ac.il

Chen Meiri
Affiliation: Einstein Institute of Mathematics, Hebrew University, Jerusalem 90914, Israel
Address at time of publication: Institute for Advanced Study, Princeton, New Jersey 08540
Email: chen7meiri@gmail.com

DOI: https://doi.org/10.1090/S0894-0347-2012-00736-X
Keywords: Sieve, property-$𝜏$, powers, linear groups, finite groups of Lie type
Received by editor(s): July 19, 2011
Received by editor(s) in revised form: January 20, 2012
Published electronically: April 11, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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