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 if $|S|\ge 2$, then every unimodular $(2\times 2)$-matrix over ${\mathcal{O}}_S$ is a product of $\le 7$ elementary matrices. This nearly optimal bound essentially concludes the investigation of bounded elementary generation of $S{L}_2({\mathcal{O}}_S)$ started over 50 years ago.

20 pages. v2: minor corrections