Bounded generation of SL 2 over rings of S-integers with infinitely many units
Algebra and Number Theory
Let be the ring of S-integers in a number field k. We prove that if the group of units × 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.
Morgan, Aleksander V.; Rapinchuk, Andrei S.; and Sury, Balasubramanian, "Bounded generation of SL 2 over rings of S-integers with infinitely many units" (2018). Journal Articles. 1532.