Article
Generalized Commutator Formulas
Communications in Algebra (Impact Factor: 0.39). 04/2011; 39(4):14411454. DOI: 10.1080/00927871003738964
ABSTRACT
Let A be an algebra which is a direct limit of module finite algebras over a commutative ring R with 1. Let I, J be twosided ideals of A, GL(n)(A, I) the principal congruence subgroup of level I in GL(n)(A), and E(n)(A, I) the relative elementary subgroup of level I. Using Bak's localizationpatching method, we prove the commutator formula [E(n)(A, I), GL(n)(A, J)] = [E(n)(A, I), E(n)(A, J)], which is a generalization of the standard commutator formular. This answers a problem posed by Stepanov and Vavilov.
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 "This was done by L. Vaserstein in [37]. Relative commutator formula was obtained by different methods in [39] [40] [17] [15]. Relative multicommutator formula for G = GL n was recently proved by R. Hazrat and Zhang Zuhong in [18]. "
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ABSTRACT: The article contains a survey of the recent author’s results on the structure of a Chevalley group G(R) over a ring R. They generalize and improve previously known results on: (1) the relative localglobal principle; (2) generators of a relative elementary group; (3) relative multicommutator formulas; (4) the nilpotent structure of a relative K1; (5) the bounded length of commutators. The proof of the first two items is based on computations with generators of the elementary group translated into the language of parabolic subgroups. The other results are proved by means of enlarging a relative elementary group, constructing a generic element, and using the localization procedure in the universal ring. 
 "The reason is that the first of these choices is too small as the neighbourhood on the right hand side, while the second of these choices is too large as the neighbourhood on the left hand side. The solution proposed for GL(n, R) in [33] and later applied to unitary groups in [32] consists in selecting another base of neightborhoods E(Φ, s k a) ≤ E(Φ, s k R, s k a) ≤ E(Φ, R, s k a), which is much better balanced with respect to conjugation. The following definition embodies the gist of this method. "
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ABSTRACT: We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions of the conjugation calculus and the commutator calculus in Chevalley groups $G(\Phi,R)$, $\rk(\Phi)\geq 2$, which are both more general, and substantially easier than the ones available in the literature. For classical groups such relative commutator calculus has been recently developed by the authors in \cite{RZ,RNZ}. As an application we prove the mixed commutator formula, \[ \big [E(\Phi,R,\ma),C(\Phi,R,\mb)\big ]=\big [E(\Phi,R,\ma),E(\Phi,R,\mb)\big], \] for two ideals $\ma,\mb\unlhd R$. This answers a problem posed in a paper by Alexei Stepanov and the second author. 
 "In fact, localisation methods used in the proof of these results have many further applications, both actual and potential: relative commutator formulas, multiple commutator formulas, nilpotency of K 1 , description of subnormal subgroups, description of various classes of overgroups, connection with excision kernels, etc. We refer to our surveys [36] [31] [32] and to our papers [29] [35] [7] [40] [37] [38] [41] [76] [39] [33] [34] for these and further applications and many further related references. "
Article: Commutator width in Chevalley groups
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ABSTRACT: The present paper is the [slightly expanded] text of our talk at the Conference "Advances in Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert that [elementary] Chevalley groups very rarely have finite commutator width. The reason is that they have very few commutators, in fact, commutators have finite width in elementary generators. We discuss also the background, bounded elementary generation, methods of proof, relative analogues of these results, some positive results, and possible generalisations.