Unitary Groups over General Classes of Form Rings

Abstract

The problems considered in this chapter are the unitary analogues of those considered in Chapter 4 for the linear groups. For the most part they have their origins in the classical questions studied in Chapter 6 over division rings. Among the questions which we will consider, will be the following: Can the normal subgroups of a unitary group U(M) be classified? Is there a stability theory for the unitary K₁ and K₂? Does an analogue of the congruence subgroup property hold for SU(M)? Are the groups U(M) or its important subgroups finitely generated or finitely presented? These problems have been an important driving force in the development of the theory of the unitary groups over rings. They are all deep and a rich and extensive literature represents the contributions made to them. It will be our strategy to select the most general results with the proviso that we will streamline whenever these are excessively technical. We will give some historical perspectives and we will limit ourselves for the most part only to brief indications of the proofs. Fuller details are provided (see §9.1 A for example) when we thought it necessary to connect a particular discussion with related material from earlier in the book. Some of the results which we will discuss are proved in the literature in the context of Chevalley groups. Refer to the Concluding Remarks of this book for a brief introduction to these groups and some indication as to how they are related to the unitary group of a quadratic module.