Abstract:
The classical work of Bass and Milnor on "lower algebraic K-theory" attracted major attention to the following questions, among others:

1. Normality of the elementary subgroup E(n,R) in the general linear group GL(n,R);
2. Classification of subgroups in GL(n,R), normalised by E(n,R);
3. Centrality of the extension St(n,R) --> E(n,R).

There are the so called STANDARD answers to these questions, which mimic the behaviour of these groups over fields and at the stable level. These standard answers are known as the MAIN STRUCTURE THEOREMS and there are several methods to prove these results, using geometry, K-theory or ring theory in various proportions.

In the present talk (based on a joint paper with Alexei Stepanov, to appear in `K-theory') we present new purely geometric proofs of the main structure theorems and some related results (like, say, description of certain classes of subgroups in GL(n,R)) for the case of a commutative ring, based on the method we call DECOMPOSITION OF UNIPOTENTS. Even for the case of fields these proofs are much simpler than the proofs of results like Jordan-Dickson theorem, usually reproduced in textbooks.

We discuss also some generalisations of these results to non-commutative rings (a very challenging and not fully solved problem is that of an exact class of rings when the standard answers hold) and mention generalisation to other groups and modules (based on our joint works with Eugene Plotkin, Anthony Bak, Elena Perelman and Michail Gavrilovich).