## Is there a finite-index finite-depth II_1 subfactor which is more than 7-super-transitive?

Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar.

There are plenty of examples of 3-super-transitive (3-ST) subfactors; Haagerup, S_4 < S_5, and others. There's exactly one known example of a 5-ST subfactor, the Haagerup-Asaeda subfactor, and one 7-ST subfactor, the extended Haagerup subfactor.

Below index 4 there are the $A_n$ and $D_n$ families, which are arbitrarily super-transitive. Ignore those; I'm just interested above index 4.

Is there anything that's even more super-transitive?

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(Aside: I'm not asking this in the expectation that anyone will have an immediate answer --- if you have one, you should answer here and then hurry off and submit to a journal!) I'm mostly asking here to record a bet with Emily Peters: if such exists, I owe her a bottle of champagne. If you can prove that none exist, she owes me a bottle of champagne. I win by default on my 60th birthday. – Scott Morrison Oct 7 2009 at 20:39