Bounded generation of $\mathrm{SL}_2$ over rings of $S$-integers with infinitely many units

Abstract

Let $\mathcal{O}$ be the ring of $S$-integers in a number field $k$. We prove that if the group of units ${\mathcal{O}}^{\times }$ is infinite then every matrix in $\mathrm{\Gamma }={SL}_2\left(\mathcal{O}\right)$ is a product of at most 9 elementary matrices. This essentially completes a long line of research in this direction. As a consequence, we obtain a new proof of the fact that $\mathrm{\Gamma }$ is boundedly generated as an abstract group that uses only standard results from algebraic number theory.