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 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 Γ = SL2(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