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

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 is a finitely generated non-virtually solvable linear group of characteristic zero, then the set of proper powers in 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:
http://dx.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.