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Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Is $\text{SL}_2(\mathcal{O})$ generated by elementary matrices? If it isn't, is there any other natural generating set for it?

The usual argument shows that this is true for $\mathcal{O} = \mathbb{Z}$ (or, more generally, a Euclidean domain). However, I haven't been able to generalize this to other rings of integers.

flag	asked Aug 29 at 16:19 Sue 149●1●7

See uni-math.gwdg.de/nica/useiqr.pdf. – Guntram Aug 29 at 16:53

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@Guntram : OK, reading that article it looks like it follows from results of Cohn and Vaserstein that it is so generated if and only if $\mathcal{O}$ is not a non-Euclidean ring of integers in an imaginary quadratic field. Why did you post this in the comments instead of as an answer? – Sue Aug 29 at 16:55

@Sue: this is not the statement. If you read Nica's survey more carefully, you see that in the case of an imaginary quadratic field, in a few exceptional cases, there is elementary generation. – Yves Cornulier Aug 30 at 6:28

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@Yves Cornulier : It is the right statement. Observe the phrase non-Euclidean in my comment. If you read Nica's survey even more carefully, you'd see that the imaginary quadratic fields for which there is elementary generation are exactly the Euclidean ones. – Sue Aug 30 at 16:13

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@Sue: sorry, you're right. I got confused between $Z[\sqrt{-d}]$ and the ring of integers. Anyway, as an answer to your first comment: you can write an answer based on the Nica's survey (Guntram's link) and put it Community Wiki. (Such a short answer as "see (link)" is not perennial insofar as the link can disappear.) – Yves Cornulier Aug 30 at 19:48

Generators for SL_2(R) for rings of integers R

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