The yoga of commutators : further applications

In the present paper, we describe some recent applications of localization methods to the study of commutators in the groups of points of algebraic and algebraic-like groups, such as GL(n,R), Bak's unitary groups GU(2n,R, Λ), and Chevalley groups G(Φ,R). In particular, we announce the multiple relative commutator formula and the general multiple relative commutator formula, as well as results on the bounded width of relative commutators in the elementary generators. We also state some of the intermediate results, as well as some corollaries of these results. At the end of the paper we attach an updated list of unsolved problems in the field.