On a question of M. Newman on the number of commutators (Q1105701)

Let F be a field of infinite transcendence degree (e.g. \({\mathbb{R}}\) or \({\mathbb{C}})\) and \(n\geq 2\) an integer. The authors show that for every integer c there is a matrix in the group SL(n,F[x]) which cannot be written as a product of c commutators. F[x] is a Euclidean domain, so for \(n\geq 3\), SL(n,F[x]) is its own commutator group. This answers negatively a question of M. Newman, who suggested the possibility that such Euclidean rings do not exist. Surprizingly the authors then show, for example, that if R is a Euclidean domain in which for some integer c every element of SL(n,R) is a product of at most c commutators, for all n large and at least 3, then in fact each such element is a product of at most 6 commutators. The authors have other positive results of this type.