Adyan, S. I.; Mennicke, J. On bounded generation of \(\text{SL}_n(\mathbb Z)\). (English) Zbl 0794.20061 Int. J. Algebra Comput. 2, No. 4, 357-365 (1992). Let \(O\) be the ring of integers of an algebraic number field and let \(A\in\text{SL}_ n(O)\), where \(n\geq 3\). By a classical theorem of Bass, Milnor and Serre \(A\) is a product of elementary matrices. Let \(\nu_ n(A)\) denote the smallest number of elementary matrices in such a product. D. Carter and G. Keller [Am. J. Math. 105, No. 3, 673–687 (1983; Zbl 0525.20029)] have obtained an upper bound for \(\nu_ n(A)\) which depends only on \(n\) and the discriminant of \(O\). Their proof involves fairly heavy machinery from algebraic number theory, including some class field theory. For the special case \(O=\mathbb Z\) they proved that \(\nu_ n(A)\leq {1\over 2}(3n^ 2-n)+ 51\). In the paper under review the authors, using a more elementary approach, prove that, for the case \(O=\mathbb Z\), \(\nu_ n(A)\leq {1\over 2}(3n^ 2-n) +61\). Reviewer: A.W.Mason (Glasgow) Cited in 14 Documents MSC: 20H05 Unimodular groups, congruence subgroups (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups 20G30 Linear algebraic groups over global fields and their integers 20F24 FC-groups and their generalizations Keywords:special linear group; bounded generation; ring of integers; algebraic number field; product of elementary matrices Citations:Zbl 0525.20029 PDFBibTeX XMLCite \textit{S. I. Adyan} and \textit{J. Mennicke}, Int. J. Algebra Comput. 2, No. 4, 357--365 (1992; Zbl 0794.20061) Full Text: DOI