2005/03/31 by Dave Witte Morris · 1 citation
Mathematics · #math.GR #math.KT #math.NT #msc:20H05 #msc:11F06 #msc:19B37
published as New York Journal of Mathematics 13 (2007) 383-421; http://nyjm.albany.edu/j/2007/13-17.html · 44 pages, no figures. Many minor errors corrected
arxiv created 2007/09/17 · arxiv updated 2009/12/01
We present unpublished work of D.Carter, G.Keller, and E.Paige on bounded generation in special linear groups.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.) 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 is in 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 > 2, these results remain valid when SL(n,A) is replaced by any of its subgroups of finite index.