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On bounded generation of \(\text{SL}_n(\mathbb Z)\). (English) Zbl 0794.20061

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 \(O=\mathbb Z\), \(\nu_ n(A)\leq {1\over 2}(3n^ 2-n) +61\).

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

Citations:

Zbl 0525.20029
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