Unimodular commutators

Abstract:

Let $R$ be a principal ideal ring and ${M_{k,n}}$ the set of $k \times n$ matrices over $R$. The following statments are proved: (a) If $k \leq n/3$ then any primitive element of ${M_{k,n}}$ occurs as the first $k$ rows of the commutator of two elements of ${\text {SL(}}n,R{\text {)}}$. (b) If every element of ${\text {SL(}}3,R{\text {)}}$ is the product of at most ${c_3}$ commutators, then every element of ${\text {SL(}}n,R{\text {)}}$ is the product of at most ${c_n}$ commutators, where ${c_n} < c\log n + {c_3} - 3,c = 2\log (3/2) = 4.932 \ldots$, and $n \geq 3$. (c) If $n \geq 3$, then every element of ${\text {SL(}}n,Z{\text {)}}$ is the product of at most $c\log n + 40$ commutators, where $c$ is given in (b) above