Sieve Methods in Group Theory I: Powers in Linear Groups

The sieve method is a classic one in number theory (see, for example, [FI]). Recently it found some applications in a non-commutative setting. On the one hand, Bourgain-Gamburd-Sarnak [BGS1] applied it in studying almost-prime vectors in orbits of non-commutative groups acting on Z. On the other hand, Rivin [Ri] and Kowalski [Ko] used it to study generic properties of elements in the mapping class group and arithmetic groups. Our formulation of the sieve method generalizes and simplifies the second one and usually falls under the name ‘Large Sieve’. The goal of this introduction is to state the general large sieve setting with respect to group theory and to serve as a guideline for the proof of Theorem A below. We start by describing the algebraic problem and its background. After that, we explain how random walks and sieve methods are used in its solution. A virtually nilpotent group is a group which contains a nilpotent subgroup of finite index. Mal′cev proved: