Width of SL(n,O_S,I)

TL;DR

This paper estimates the width of certain congruence subgroups of SL(n,O_S,I) in Tits--Vaserstein generators, considering number fields with specific properties, contributing to understanding their algebraic structure.

Contribution

It provides new bounds for the width of SL(n,O_S,I) in Tits--Vaserstein generators under specific conditions on the number field and ideal.

Findings

Established bounds for the width of SL(n,O_S,I)

Extended results to number fields with real embeddings

Analyzed cases where I is prime to roots of unity

Abstract

We give an estimate for the width of the congruence subgroup $\mathrm{SL}(n,O_S,I)$ in Tits--Vaserstein generators, where $O_S$ is a localisation of the ring of integers in a number field $K$. We assume that either $K$ has a real embedding, or the ideal $I$ is prime to the number of roots of unity in $K$.