Sieving and expanders^**The first author was supported in part by NSF grant DMS-0322370. The second author was supported in part by NSF grant DMS-0111298 and DMS-0501245. The third author was supported in part by Oscar Veblen Fund (IAS) and the NSF.

Let V be an orbit in Zⁿ of a finitely generated subgroup Λ of GL_n (Z) whose Zariski closure Zcl (Λ) is suitably large (e.g. isomorphic to SL₂). We develop a Brun combinatorial sieve for estimating the number of points on V for which a fixed set of integral polynomials take prime or almost prime values. A crucial role is played by the expansion property of the 'congruence graphs' that we associate with V. This expansion property is established when Zcl (Λ) = SL₂. To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).