Sieves in discrete groups, especially sparse

Summary

We survey the recent applications and developments of sieve methods related to discrete groups, especially in the case of infinite index subgroups of arithmetic groups.

Sieve methods appeared in number theory as a tool to try to understand the additive properties of prime numbers, and then evolved over the twentieth century into very sophisticated tools. Not only did they provide extremely strong results concerning the problems most directly relevant to their origin (such as Goldbach’s conjecture, the twin primes conjecture, or the problemof the existence of infinitely many primes of the form n²+1). but they also became tools of crucial important in the solution of many problems which were not so obviously related (examples are the first proof of the Erdős–Kac theorem, and more recently sieve appeared in the progress, and solution, of the quantum unique ergodicity conjecture of Rudnick and Sarnak).

It is only quite recently that sieve methods have been applied to new problems, often obviously related to the historical roots of sieve, which involve complicated infinite discrete groups (of exponential growth) as basic substrate instead of the usual integers. Moreover, both “small” and “large” sieves turn out to be applicable in this context to a wide variety of very appealing questions, some of which are rather surprising. We will attempt to present this story in this survey.