Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|cite|improve this question
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

1 Answer 1

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 $k$.

share|cite|improve this answer
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 at 2:07

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.