Bounded generation of SL ( n , A ) ( after D . Carter ,

We present unpublished work of D. Carter, G. Keller, and E. Paige on bounded generation in special linear groups.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 of its subgroups of finite index.