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

TL;DR

This paper discusses the bounded generation property of special linear groups over rings of integers in algebraic number fields, showing that certain subgroups can be generated by a bounded number of elementary matrices or conjugates.

Contribution

It proves bounded generation results for SL(n,A) and its subgroups, extending previous work to more general rings and subgroups of finite index.

Findings

Existence of finite-index subgroups with bounded elementary generation

Bounded generation by conjugates of non-scalar matrices in SL(n,A)

Results hold for n > 2 and for various subgroups of SL(n,A)

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We present unpublished work of D.Carter, G.Keller, and E.Paige on bounded generation in special linear groups.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