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On a question of M. Newman on the number of commutators. (English) Zbl 0649.20048

Let F be a field of infinite transcendence degree (e.g. \({\mathbb{R}}\) or \({\mathbb{C}})\) and \(n\geq 2\) an integer. The authors show that for every integer c there is a matrix in the group SL(n,F[x]) which cannot be written as a product of c commutators. F[x] is a Euclidean domain, so for \(n\geq 3\), SL(n,F[x]) is its own commutator group. This answers negatively a question of M. Newman, who suggested the possibility that such Euclidean rings do not exist.
Surprizingly the authors then show, for example, that if R is a Euclidean domain in which for some integer c every element of SL(n,R) is a product of at most c commutators, for all n large and at least 3, then in fact each such element is a product of at most 6 commutators. The authors have other positive results of this type.
Reviewer: B.A.F.Wehrfritz

MSC:

20H25 Other matrix groups over rings
20F12 Commutator calculus
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[1] Bass, H., \(k\)-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math., 22, 5-60 (1964) · Zbl 0248.18025
[2] Brown, L. G.; Schochet, C., \(K_1\) of the compact operators is zero, (Proc. Amer. Math. Soc., 59 (1976)), 119-122 · Zbl 0345.47024
[3] Carter, D.; Keller, G., Bounded elementary generation of (), Amer. J. Math., 105, 673-687 (1983) · Zbl 0525.20029
[4] Carter, D.; Keller, G., Elementary expressions for unimodular matrices, Comm. Algebra, 12, 379-389 (1984) · Zbl 0572.20030
[5] Carter, D.; Keller, G., Bounded elementary expressions in (2, ) (1985), preprint
[6] Carter, D.; Keller, G., The congruence subgroup problem for nonstandard models (1985), preprint
[7] Carter, D.; Keller, G.; Paige, E., Bounded expressions in (2, ) (1985), preprint
[8] Cohn, P. M., Determinants on free fields, Contemp. Math., 13, 99-108 (1982) · Zbl 0505.16001
[9] Cooke, G. E.; Weinberger, P. J., On the construction of division chains in algebraic number rings, with applications to \(SL_2\), Comm. Algebra, 3, 481-524 (1975) · Zbl 0315.12001
[10] R. K. Dennis and L. N. Vaserstein; R. K. Dennis and L. N. Vaserstein · Zbl 0676.20024
[11] Draxl, P., Eine Liftung der Dieudonné-Determinante und Anwendungen die Multiplicative Gruppe eines Schiefkörpers betreffend, (Lecture Notes in Mathematics, Vol. 778 (1980), Springer-Verlag: Springer-Verlag Berlin/New York), 101-116
[12] Gupta, R.; Murty, M. Ram, A remark on Artin’s conjecture, Invent. Math., 78, 127-130 (1984) · Zbl 0549.10037
[13] Gupta, R.; Murty, M. Ram; Murty, V. Kumar, Artin’s conjecture and the Euclidean algorithm (1985), preprint
[14] de la Harpe, P.; Skandalis, G., Déterminant associé à une trace sur une algèbre de Banach, Ann. Inst. Fourier (Grenoble), 34, No. 1, 241-260 (1984) · Zbl 0521.46037
[15] Kursov, V. V., The commutant of the general linear group over field, Dokl. Akad. Nauk BSSR, 23, No. 10, 869-871 (1979) · Zbl 0415.20028
[16] Kursov, V. V., Commutators of the multiplicative group of a finite dimensional central division algebra, Dokl. Akad. Nauk BSSR, 26, No. 2, 101-103 (1982) · Zbl 0498.16017
[17] Lenstra, H. W., On Artin’s conjecture and Euclid’s algorithm in global fields, Invent. Math., 42, 201-224 (1971) · Zbl 0362.12012
[18] Liehl, B., Beschränkte Wortlänge in \(SL_2\), Math. Z., 186, 509-524 (1984) · Zbl 0523.20028
[19] Newman, M., Matrix completion theorems, (Proc. Amer. Math. Soc., 94 (1985)), 39-45 · Zbl 0564.15007
[20] Rehmann, U., Kommutatoren in \(GL_n(D)\), (Lecture Notes in Mathematics, Vol. 778 (1980), Springer-Verlag: Springer-Verlag Berlin/New York), 117-123
[21] Thompson, R. C., Commutators in the special linear and general linear groups, Trans. Amer. Math. Soc., 101, 16-33 (1961) · Zbl 0109.26002
[22] van der Kallen, W., \( SL_3(C\)[x]) does not have bounded word length, (Lecture Notes in Mathematics, Vol. 966 (1982), Springer-Verlag: Springer-Verlag Berlin/New York), 357-361 · Zbl 0935.20501
[23] Vaserstein, L. N., Math. Notes, 5, 141-148 (1969) · Zbl 0195.32202
[24] Vaserstein, L. N., Functional Anal. Appl., 5, 102-110 (1971) · Zbl 0239.16028
[25] Vaserstein, L. N., Bass’s first stable range condition, J. Pure Appl. Algebra, 34, 319-330 (1984) · Zbl 0547.16017
[26] Wood, J. W., Bundles with totally disconnected structure group, Comment. Math. Helv., 46, 257-273 (1971) · Zbl 0217.49202
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