On the decomposition of automorphisms of free modules into simple factors

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.