New York Journal of Mathematics
Volume 13 (2007) 383-421


Dave Witte Morris

Bounded generation of SL(n,A) (after D. Carter, G. Keller, and E. Paige)

Published: September 27, 2007
Keywords: Bounded generation, finite width, special linear group, elementary matrix, stable range, Mennicke symbol, nonstandard analysis
Subject: 20H05; 11F06, 19B37

We present unpublished work of D. Carter, G. Keller, and E. Paige on bounded generation in special linear groups.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


Partially supported by a grant from the National Sciences and Engineering Research Council of Canada.

Author information

Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada