On the bounded generation of arithmetic SL<sub>2</sub>.
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- Record sourced from PubMed, PMID 31484782.
- Also identified by DOI 10.1073/pnas.1907728116 and PMC identifier 6754582.
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Abstract
Let <i>K</i> be a number field and <i>S</i> be a finite set of primes of <i>K</i> containing the archimedean valuations. Let 𝒪 be the ring of <i>S</i>-integers in <i>K</i> Morgan, Rapinchuck, and Sury [A. V. Morgan <i>et al</i>, <i>Algebra Number Theory</i> 12, 1949-1974 (2018)] have proved that if the group of units [Formula: see text] is infinite, then every matrix in SL<sub>2</sub>(𝒪) is a product of at most 9 elementary matrices. We prove that under the additional hypothesis that <i>K</i> has at least 1 real embedding or <i>S</i> contains a finite place we can get a product of at most 8 elementary matrices. If we assume a suitable generalized Riemann hypothesis, then every matrix in SL<sub>2</sub>(𝒪) is the product of at most 5 elementary matrices if <i>K</i> has at least 1 real embedding, the product of at most 6 elementary matrices if <i>S</i> contains a finite place, and the product of at most 7 elementary matrices in general.