 

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 


Abstract
We present unpublished work of D. Carter, G. Keller, and E. Paige on bounded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 finiteindex 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
finiteindex, 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.


Acknowledgements
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
Dave.Morris@uleth.ca
http://people.uleth.ca/~dave.morris/

