Commutators in special linear groups over certain division rings
Abstract
UDC 512.5
We consider the question whether an element of a special linear group ${\rm SL}_m(D)$ of degree $m\ge 1$ over a division ring $D$ is a commutator. Our first aim is to show that if the division ring $D$ is algebraically closed and finite-dimensional over its center, then every element of ${\rm SL}_m(D)$ is a commutator of ${\rm SL}_m(D).$ We also indicate that this question is related to the derived series in division rings and then describe the derived series in the Mal'cev–Neumann division rings of noncyclic free groups over fields.
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