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Edit: I'm specializing this question to the compact case I'll ask about the noncompact case as a new question.

Let $ G $ be a compact connected semisimple Lie group. Do there always exist two finite order elements of $ G $ which generate a dense subgroup?

Example: $$ \frac{i}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} $$ and $$ \begin{bmatrix} \overline{\zeta_{16}} & 0 \\ 0 & \zeta_{16} \end{bmatrix} $$ generate a dense subgroup of $ SU_2 $.

This question is partially inspired by

Generating finite simple groups with $2$ elements

which shows that every finite simple group is 2-generated. Indeed even every finite quasisimple group is 2-generated Is every finite quasi-simple group generated by 2 elements? (however this fails for infinite simple groups: $ \mathrm{PSL}_2(\mathbb{Q}) $ is not 2-generated indeed not even finitely generated).

This is a cross-post of https://math.stackexchange.com/questions/4537024/dense-subgroups-generated-by-two-finite-order-elements an answer posted there points out that every compact semisimple Lie group can be topologically generated by 2 infinite order elements.

Ian Gershon Teixeira
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    I think that there's a paper that proves this by some mechanism like showing the set of pairs that generate has positive Haar measure. I'll see if I can dig it up. – LSpice Sep 29 '22 at 15:25
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    Theorem 2.1 in this J. Algebra paper of Breuillard-Gelander should help: indeed it is then enough to show that every (connected, finite center) semisimple Lie group is (topologically) generated by two circle subgroups. – YCor Sep 29 '22 at 16:10
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    [Finite center should be assumed. Or, at least one has to exclude the case when $G$ surjects onto the universal cover of $\mathrm{SL}_2(\mathbf{R})$, which is torsion-free.] – YCor Sep 29 '22 at 16:11
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    Good point I've added the assumption of finite center – Ian Gershon Teixeira Apr 10 '23 at 14:46
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    You should add the assumption that $G$ is semisimple, otherwise the claim is clearly false since $G$ can be torsion-free, e.g. the subgroup $B< SL(2,{\mathbb R})$ of non-strictly upper triangular matrices with positive diagonal entries. – Moishe Kohan Apr 10 '23 at 14:57
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    I still haven't dug up the paper I was thinking of, but this question may be relevant: Word maps on compact Lie groups (and maybe also the assumption of compactness was in the paper that I remembered). – LSpice Apr 10 '23 at 16:47

1 Answers1

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You might want to look at MR0034766 (Kuranishi 1949, link). It deals with the connected semisimple case, which might be what you're after.

There's also a 1999 Proc AMS paper by M. Field, (freely accessible at AMS site — MR number for subscribers: MR1618662), which shows that the set of pairs generating a dense open subset of $G$ is non-empty Zariski open in $G \times G$ when $G$ is compact connected semisimple. This should lead to a proof that the elements can be taken to have finite order in this case, as the set of pairs of elements of finite order is dense in $G\times G$. Of course, in the non-compact case some extra condition on $G$ is required.

YCor
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Dave Benson
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    The MR link for this is erronious. The paper itself is here: https://projecteuclid.org/journals/kodai-mathematical-journal/volume-1/issue-5-6/Two-elements-generations-on-semi-simple-Lie-groups/10.2996/kmj/1138833534.full The compact case is due to Auerbach. – Dave Benson Apr 12 '23 at 19:21
  • I'm not sure what you mean by the MathReviews link being erroneous. This looks correct to me: https://mathscinet.ams.org/mathscinet-getitem?mr=34766 – Sam Hopkins Apr 12 '23 at 19:26
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    What I mean is that it the "article" link is to the wrong paper. It links to a paper by Kawata. – Dave Benson Apr 12 '23 at 19:27
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    @DaveBenson, there's a Contact Us link on the MathSciNet page, that I've used to good effect when dealing with bad MR links. I have just reported the error you mentioned, with credit to you. The correct DOI is 10.2996/kmj/1138833534. For your other reference, MR1618662, the DOI is 10.1090/S0002-9939-99-04959-X. – LSpice Apr 12 '23 at 21:31
  • Thanks, @LSpice. I've sometimes done that, but I come across them often enough that it's annoying to have to do it every time. – Dave Benson Apr 12 '23 at 21:33
  • If $G$ is semisimple non-compact, the set of finite order elements is never dense (it can have empty or non-empty interior: this depends on $G$). – YCor Apr 13 '23 at 00:14
  • So for $ G $ compact connected semisimple then we have 1) The pairs of finite order elements are Zariski dense in $ G \times G $ 2) The pairs that topologically (with respect to the manifold topology) generate $ G $ are non-empty Zariski open in $ G \times G $ 3) The intersection of non-empty open with dense is non-empty ( the intersection should even be infinite since it is dense in the non-empty open set). So for a compact connected semisimple Lie group $ G $ there are infinitely many pairs $ (g,h) $ with $ g $ and $ h $ finite order and $ \langle g,h \rangle $ (Euclidean) dense in $ G $. – Ian Gershon Teixeira Apr 13 '23 at 13:57
  • Also I'm inclined to just award the bounty and then create a new question for the non-compact case but I'll wait a day and see if anything else interesting can be said. – Ian Gershon Teixeira Apr 13 '23 at 14:00
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    @IanGershonTeixeira Yes, exactly. This answers your question for compact connected semisimple, but not for noncompact connected semisimple with finite centre, where I don't know the answer. – Dave Benson Apr 13 '23 at 17:05
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    @DaveBenson Thanks for pointing out the error. We will look into the erroneous article link. We get many of the links from the DOI repository, CrossRef. Lately, we seem to have had a higher number of bad links of this sort. We aren't sure if it's our matching algorithm or CrossRef's data. – Edward Dunne Apr 14 '23 at 19:02
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    Cheers, Ed! As always, thanks for your help. MathSciNet is a fantastically useful resource. – Dave Benson Apr 14 '23 at 19:06
  • alright I'm just going to restrict this question to the compact case since this is the accepted answer and make a new question for the noncompact case. – Ian Gershon Teixeira Apr 18 '23 at 13:52