Abstract

Localisation is one of the most powerful ideas in the study of [the groups of points of] algebraic-like groups, and classical groups. It was first developed in this context by Daniel Quillen and Andrei Suslin in 1976-77, in their solution of Serre's problem and its higher analogues, and further applied by Vyacheslav Kopeiko, Leonid Vaserstein, Eiichi Abe, and others. In 1990 Anthony Bak came up with another version of localisation, localisation-completion. Some similar methods were further developed by Ravi Rao and his students Rabeya Basu, Selby Jose, and by others.
In the present talk we describe three recent versions of localisation methods:
-- Relative localisation,
-- Universal localisation,
-- Localisation completion,

and show some typical applications of these methods, like:
-- Relative commutator formulae,
-- Bounded width of commutators in terms of elementary generators,
-- Nilpotent structure of [relative] K_1.

All localisation methods rely on a large common body of elementary, but very strenuous calculations, known as the yoga of conjugation, and the yoga of commutators, and we will try to convey also some feel of those.




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