Width of SL(n,O_S,I)

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 $K$.