Bounded elementary generation of SL_2: nearly the end

Abstract

Let $\mathcal{O}_S$ be the ring of $S$-integers of a global field $K$ of any characteristic, where $S$ is a finite set of valuations of $K$ (and $S$ contains all of the archimedean valuations if the characteristic is zero). We prove that every unimodular $(2 \times 2)$-matrix over $\mathcal{O}_S$ is a product of $\leq 7$ elementary matrices. This nearly optimal bound essentially concludes the investigation of bounded elementary generation of $SL_2(\mathcal{O}_S)$ started over 50 years ago.