The Congruence Subgroup problem

The great mathematician Carl Friedrich Gauss was once told about a contemporary of his who was stuck on a problem due to a ‘lack of sufficient notation’. Gauss is said to have quipped that what he needed was notions — not notations. It was, indeed, Gauss who propounded the notion of congruences (and its notation ≡ too!).

This chapter proves the first non-trivial theorem of the subject which is due to Chevalley. Chevalley’s theorem solves the congruence subgroup problem for the multiplicative group of a global field. We follow this up with a generalisation due to Wehrfritz. Finally, it is shown how to use Chevalley’s theorem to solve the congruence subgroup problem for the group of upper triangular, invertible matrices.

In the previous chapter, we dealt with solvable groups and the methods came from abelian groups. In this chapter, we deal with \(S{L_2}({\mathcal{O}_S})\) where \(\mathcal{O}\) is the ring of integers in an algebraic number field and S is any set of places containing all the archimedean ones. The methods here will be very different from those of the previous chapter.

This chapter, along with the next one on applications, is the most substantial part of this book. As our aim is to have an elementary treatment, we tread a bit cautiously and take a circuitous path in some places. Starting with the description of Steinberg’s commutator relations and of the Steinberg group following the classic [Ste1], we prove that the Steinberg group gives all the central extensions of SL(n, K) for a field K. These ideas are followed up to provide a presentation of the group SL(n, ℤ/r) for any n ≥ 3 and any r ≥ 2 as well as a presentation for SL(n, ℤ) itself for n ≥ 3. This leads to our first proof of the CSP (recall the definition of the CSP given in 3–7.1) for SL(n, ℤ) for n ≥ 3. This proof is due to Steinberg ([Ste3]) and is simpler than the original proofs due (independently) to Bass, Lazard & Serre [BLS] and [Me1]. The proof uses some results of Brenner [Br] which are proved here as well.

This chapter contains a number of group-theoretic and number-theoretic applications of the congruence subgroup property. These applications are generally unrelated to each other otherwise. Let us briefly describe them.

In this chapter, we recall the notions of linear algebraic groups and their arithmetic subgroups. After developing some background material on these notions, we shall formulate the CSP in the general situation. This will be followed by a survey of the known results with regard to the CSP. This chapter is merely meant to be a brief survey of the state-of-the-art of this topic. Most of the results stated and proved for SL _n in the previous five chapters will have analogues which are either known or conjectured to be true in the existent literature. We shall hardly give any proofs except for very brief sketches of some important results of particular relevance to us like the proof of centrality of the congruence kernel. The informed reader is referred to the surveys in [PR2], last chapter and [R3].