Bounded generation of SL2 over rings of S-integers with infinitely many units

Morgan, Aleksander; Rapinchuk, Andrei; Sury, Balasubramanian

doi:10.2140/ant.2018.12.1949

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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

DOI: 10.2140/ant.2018.12.1949

Abstract

Let $O$ be the ring of $S$ -integers in a number field $k$ . We prove that if the group of units $O^{\times}$ is infinite then every matrix in $Γ = {SL}_{2} (O)$ 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 $Γ$ 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

Milestones

Received: 3 September 2017

Revised: 10 May 2018

Accepted: 18 June 2018

Published: 4 December 2018

Authors

Aleksander V. Morgan
Department of Mathematics University of Virginia Charlottesville, VA United States

Andrei S. Rapinchuk
Department of Mathematics University of Virginia Charlottesville, VA United States

Balasubramanian Sury
Stat-Math Unit Indian Statistical Institute Bangalore India

Vol. 12, No. 8, 2018