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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.

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See – Guntram Aug 29 '12 at 16:53
@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 '12 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. – YCor Aug 30 '12 at 6:28
@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 '12 at 16:13
@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.) – YCor Aug 30 '12 at 19:48

If $\mathcal O$ be the ring of integers in an algebraic number field, whether $SL_2(\mathcal O)$ is generated by elementary matrices depends on the field $k$:

  • If $k = \Bbb Q$, or $k = \Bbb Q(\sqrt{-D}$ for $D\in\{1,2,3,7,11\}$, then $SL_2(\mathcal O)$ is generated by elementary matrices.

  • If $k = \Bbb Q(\sqrt{-D})$ for $D$ any other squarefree integer, then $SL_2(\mathcal O)$ is not generated by the elementary matrices. However, the index of the subgroup generated by elementary matrices in $SL_2(\mathcal O)$ is finite, so we can add a finite number of non-elementary matrices to the generating set, to ensure $SL_2(\mathcal O)$ will be generated by this set.

  • For all other $k$, $SL_2(\mathcal O)$ is not only generated by elementary matrices, but a bounded number of matrices is enough to generate any matrix. The bound depends on the field $k$.

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what do you mean "the ring generated by elementary matrices"? Do you mean the group? The survey Guntram posted says the index is infinite. – Will Sawin Aug 25 '15 at 2:07

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