Sieve methods in group theory I: Powers in linear groups

Lubotzky, Alexander; Meiri, Chen

doi:10.1090/S0894-0347-2012-00736-X

Sieve methods in group theory I: Powers in linear groups
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by Alexander Lubotzky and Chen Meiri;

J. Amer. Math. Soc. 25 (2012), 1119-1148

DOI: https://doi.org/10.1090/S0894-0347-2012-00736-X

Published electronically: April 11, 2012

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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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