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

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

2005-03-05

arxiv.org/abs/math/0503083v3

New York Journal of Mathematics 13 (2007) 383-421; http://nyjm.albany.edu/j/2007/13-17.html

Mathematics

Group Theory

44 pages, no figures. Many minor errors corrected

Scientific paper

We present unpublished work of D.Carter, G.Keller, and E.Paige on bounded generation in special linear groups.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

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