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We present unpublished work of D. Carter, G. Keller, and E. Paige on bounded generation in special linear groups. Let n be a positive integer, and let A = O be the ring of integers of an algebraic number field K (or, more generally, let A be a localization OS −1). If n = 2, assume that A has infinitely many units. We show there is a finite-index subgroup H of SL(n, A), such that every matrix in H is a product of a bounded number of elementary matrices. We also show that if T ∈ SL(n, A), and T is not a scalar matrix, then there is a finite-index, normal subgroup N of SL(n, A), such that every element of N is a product of a bounded number of conjugates of T. For n ≥ 3, these results remain valid when SL(n, A) is replaced by any o

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Year: 2007

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