Strong approximation for algebraic groups

Summary

We survey results on strong approximation in algebraic groups, considering in detail the classical form of strong approximation as well as more recent results on strong approximation for arbitrary Zariski-dense subgroups. Some other topics, ranging from strong approximation in homogeneous spaces of algebraic groups to various applications of strong approximation, are also discussed.

This article is a survey of known results related to strong approximation in algebraic groups. We focus primarily on two aspects: the classical form of strong approximation, which is really strong approximation for S-arithmetic groups (Section 2), and its more modern version for arbitrary Zariski-dense subgroups (Section 3). Along the way we will also mention results dealing with strong approximation in arbitrary varieties and particularly homogeneous spaces (which are probably not so well known to the general audience as some other results in the article) and some applications. The reader will find more applications of strong approximation for Zariski-dense subgroups in other articles in this volume.