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 Z n of a finitely generated subgroup Λ of GL n ( Z ) whose Zariski closure Zcl ( Λ ) is suitably large (e.g. isomorphic to SL 2 ). 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 ) = SL 2 .