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arXiv · 2606.21817

Bounded elementary generation of SL_2: nearly the end

B. E. Kunyavskii·D. W. Morris·A. S. Rapinchuk
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Abstract

Let $\mathcal{O}_S$ be the ring of $S$-integers of a global field $K$ of any characteristic, where $S$ is a finite set of valuations of $K$ (and $S$ contains all of the archimedean valuations if the characteristic is zero). We prove that if $|S| \ge 2$, then every unimodular $(2 \times 2)$-matrix over $\mathcal{O}_S$ is a product of $\leq 7$ elementary matrices. This nearly optimal bound essentially concludes the investigation of bounded elementary generation of $SL_2(\mathcal{O}_S)$ started over 50 years ago.

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B. E. Kunyavskii, D. W. Morris, A. S. Rapinchuk. 2026-09-09. Bounded elementary generation of SL_2: nearly the end. https://arxiv.org/abs/2606.21817

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