On Quillen’s local global principle

“…It seems that most necessary tools are contained already in the works by Victor Petrov, Tony Bak, Rabeya Basu, Ravi Rao, and Reema Khanna [10,[17][18][19][20][80][81][82].…”

Section: Problem 23mentioning

confidence: 99%

The Yoga of Commutators: Further Applications

et al. 2014

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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. 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. 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:• relative Quillen-Suslin principle [7];• relative localization-completion [41].Also, we state some subsidiary results, which are not directly based on localization methods, but rather are pure group theory. Nevertheless, we believe they are fun in themselves and could be useful in further studies of commutators in algebraic-like groups. Here are three typical results of this kind for the relative elementary subgroups, which we discuss in the present paper:• generation of relative commutator subgroups [47][48][49]51].

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“…Vyacheslav Kopeiko and Andrei Suslin [29], [52], Giovanni Taddei, Leonid Vaserstein, You Hong, Anthony Bak, and the present authors generalised these results to other groups. In [6], [8] - [10], [16], [19], [22], [23], [30], [48], [64], [70] one can find different proofs of these results, many further references, and a detailed discussion of their role in the structure theory.…”

Section: Commutator Calculusmentioning

confidence: 99%

The yoga of commutators

Hazrat,

Stepanov,

Vavilov

et al. 2011

Preprint

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On Quillen’s local global principle

Cited by 19 publications

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Gram-Schmidt-Vaserstein generators for odd sized elementary groups

Gram-Schmidt-Vaserstein generators for odd sized elementary groups

The Yoga of Commutators: Further Applications

The yoga of commutators

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