Geodesic
On the decomposition of automorphisms of free modules into simple factors
Izvestiya. Mathematics , Tome 59 (1995) no. 2, pp. 333-351

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We study decompositions of automorphisms of various free modules into products of transvections and dilations. In particular, for a free $\mathbb Z$-module $M=\mathbb Z^n$ (where $n\geqslant 3$) we show that any automorphism $\sigma\in\operatorname{GL}_n(M)$ can be expressed as a product of not more than $2n+5$ transvections and one simple transformation which is a transvection if $\sigma\in\operatorname{SL}_n(M)$ and a dilation otherwise. As a corollary we obtain that for $n\geqslant 3$ the width of the group $\operatorname{SL}_n(\mathbb Z)$, with respect to the set of commutators, does not exceed 10. 
@article{IM2_1995_59_2_a4,
     author = {V. G. Bardakov},
     title = {On the decomposition of automorphisms of free modules into simple factors},
     journal = {Izvestiya. Mathematics },
     pages = {333--351},
     publisher = {mathdoc},
     volume = {59},
     number = {2},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_59_2_a4/}
}
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V. G. Bardakov. On the decomposition of automorphisms of free modules into simple factors. Izvestiya. Mathematics , Tome 59 (1995) no. 2, pp. 333-351. http://geodesic.mathdoc.fr/item/IM2_1995_59_2_a4/