Vol. 12, No. 8, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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