Combinatorics of affine birational maps

The main object of study of the present paper is the group ${\text{au}}_n$ of \emph{unimodular automorphisms} of ${\text{com}}^n$. Taking ${\text{au}}_n$ as a working example, our intention was to develop an approach (or rather an edifice) which allows one to prove, for instance, the non-simplicity of ${\text{au}}_n$ for all $n \geq 3$. More systematic and, perhaps, general exposition will appear elsewhere.