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2018 Bounded generation of $\mathrm{SL}_2$ over rings of $S$-integers with infinitely many units
Aleksander V. Morgan, Andrei S. Rapinchuk, Balasubramanian Sury
Algebra Number Theory 12(8): 1949-1974 (2018). DOI: 10.2140/ant.2018.12.1949


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.


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Aleksander V. Morgan. Andrei S. Rapinchuk. Balasubramanian Sury. "Bounded generation of $\mathrm{SL}_2$ over rings of $S$-integers with infinitely many units." Algebra Number Theory 12 (8) 1949 - 1974, 2018.


Received: 3 September 2017; Revised: 10 May 2018; Accepted: 18 June 2018; Published: 2018
First available in Project Euclid: 21 December 2018

zbMATH: 06999399
MathSciNet: MR3892969
Digital Object Identifier: 10.2140/ant.2018.12.1949

Primary: 11F06
Secondary: 11R37 , 20H05

Keywords: arithmetic groups , bounded generation , congruence subgroup problem

Rights: Copyright © 2018 Mathematical Sciences Publishers


Vol.12 • No. 8 • 2018
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