Bounded generation of SL_2 over rings of S-integers with infinitely many units

TL;DR

This paper proves that for rings of S-integers with infinitely many units, the group SL_2(O) is boundedly generated by elementary matrices, advancing understanding of its algebraic structure.

Contribution

It establishes that every element of SL_2(O) can be expressed as a product of at most 9 elementary matrices when O^* is infinite, confirming bounded generation.

Findings

Every matrix in SL_2(O) is a product of at most 9 elementary matrices.

SL_2(O) is boundedly generated as an abstract group.

The result completes a long-standing research question in the area.

Abstract

Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $\Gamma$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this direction. As a consequence, we obtain that $\Gamma$ is boundedly generated as an abstract group.