The yoga of commutators

Abstract

In the present paper we discuss some recent versions of localisation methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localisation, universal localisation, and enhanced versions of localisation-completion. Apart from the general strategic description of these methods, we state some typical technical results of the conjugation calculus and the commutator calculus. Also, we state several recent results obtained therewith, such as relative standard commutator formulae, bounded width of commutators, with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localisation methods can be much more efficient, than expected

The Yoga of Commutators: Further Applications

“…For the reader's convenience, we reproduce with minor changes the first section of [39]. As there, we consider algebraic-like or classical-like group functors G. Further, let G(R) be the group of points of G over a ring R or a form ring (R, Λ).…”

“…At the end of the paper we attach an updated list of unsolved problems in the field. Bibliography: 132 titles.The present paper is a direct continuation of [39]. Its goal is to announce some major advances we achieved in 2010-2013, after the publication of [39], with the use of the new localization methods described therein.…”

“…Its goal is to announce some major advances we achieved in 2010-2013, after the publication of [39], with the use of the new localization methods described therein. Namely, in [39] we described the methods themselves, and stated three typical applications to the study of commutators in algebraic-like groups:• relative standard commutator formulas [45,46,50,118,120];• universal length bound for commutators [42,86,92,94];• nilpotent structure of relative K 1 [9, 11, 36, 43].In the present paper, we briefly outline some further fragments of commutator calculus and state three fresh applications of these methods to the study of commutators in algebraic-like groups:• multiple commutator formula [47,49,50];• general multiple commutator formula [41,49];• relative commutator length [40,42,90].In particular, this solves several of the problems stated in [39]. Moreover, we describe two further recent relative versions of localization methods themselves:…”

“…Its goal is to announce some major advances we achieved in 2010-2013, after the publication of [39], with the use of the new localization methods described therein. Namely, in [39] we described the methods themselves, and stated three typical applications to the study of commutators in algebraic-like groups:…”

“…In particular, this solves several of the problems stated in [39]. Moreover, we describe two further recent relative versions of localization methods themselves:…”