4
$\begingroup$

The fantastic answers to my previous question Subgroups of $SL_2(\mathbb R)$ which contain $SL_2(\mathbb Z)$ as a finite index subgroup led me to the following question.

Let $O_K$ be the ring of integers of $K= \mathbb{Q}(\sqrt{p})$, where $p$ is a prime number.

What are the subgroups $\Gamma \subset \mathrm{Sp}(2g,O_K)$ of infinite index which contain $\mathrm{Sp}(2g,\mathbb Z)$? Are there any?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
6
$\begingroup$

First of all, thank you for the adjective "fantastic"(!). The question is actually studied in a paper of mine (link to the MR review). Sorry for talking about my own paper (I have no option, since I do not know if anyone else is interested enough in these questions).

[I should add that, in all this, $g\geq 2$]. In the paper referred to above, what is proved is that any intermediate subgroup either has finite index in $Sp_{2g}(O_K)$ or else contains $Sp_{2g}(\mathbb{Z})$ as a finite index subgroup. In particular, since $Sp_{2g}(\mathbb{Z})$ is a maximal discrete subgroup of $Sp_{2g}(\mathbb{R})$ by the answers to your previous question, there are no in-between subgroups of infinite index in $Sp_{2g}(O_K)$ other than $Sp_{2g}(\mathbb{Z})$.

Improve this answer
$\endgroup$
2
  • 2
    $\begingroup$ Thank you again! I have three questions if that's ok. 1) Is it important in your paper that $K$ is a real quadratic field? Or could $K$ be any number field? 2) Is it important that $O_K$ is a maximal order in $K$? What if we take any order $O$ and ask a similar question? 3) What if we look at subgroups between $Sp(2g,O_K)$ and some finite index subgroup of $Sp(2g,\mathbb Z)$? Sorry for the many questions; I'm only now discovering these fantastic results. $\endgroup$
    – Honing
    Oct 13, 2016 at 21:05
  • 1
    $\begingroup$ @Honing. Thank you ! Yes, in the paper, it was important that it was (usually real) quadratic extension. In general, it seems hard to extend the results of that paper to other extensions. Your question 3 (for real quadratic $K$) is that any intermediate group either has finite index in the larger group or else contains the smaller group as a finite index subgroup (this is weaker than what you have asked for $Sp_{2g}(\mathbb{Z})$). Yes to Question 3, in this weaker sense, implies yes to question 2. $\endgroup$
    – Venkataramana
    Oct 14, 2016 at 0:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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