Sieve methods in group theory I: Powers in linear groups

Lubotzky, Alexander ¹   ; Meiri, Chen ^{1 ,  2}

¹ Einstein Institute of Mathematics, Hebrew University, Jerusalem 90914, Israel
² Institute for Advanced Study, Princeton, New Jersey 08540

Journal of the American Mathematical Society, Tome 25 (2012) no. 4, pp. 1119-1148

Résumé

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.

Détail

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

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