#### Vol. 12, No. 8, 2018

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

### Aleksander V. Morgan, Andrei S. Rapinchuk and Balasubramanian Sury

Vol. 12 (2018), No. 8, 1949–1974
##### Abstract

Let $\mathsc{O}$ be the ring of $S$-integers in a number field $k$. We prove that if the group of units ${\mathsc{O}}^{×}$ is infinite then every matrix in $\Gamma ={SL}_{2}\left(\mathsc{O}\right)$ 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 $\Gamma$ is boundedly generated as an abstract group that uses only standard results from algebraic number theory.

##### Keywords
bounded generation, arithmetic groups, congruence subgroup problem
##### Mathematical Subject Classification 2010
Primary: 11F06
Secondary: 11R37, 20H05