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Does anyone know what the current expert consensus is concerning the status of the question as to whether the mapping class group of a surface has property (T)?

There is a short (21 page) paper by J. Andersen which purports to use quantum representations to prove that it does not. See here. It was released in 2007, but it does not seem to have yet been accepted by a journal. I have asked several experts (on the mapping class group and on property (T)), and none of them seem to understand the details of this paper. One or two of them alluded to issues they had heard might exist, but they were pretty vague as to what these issues might be.

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    $\begingroup$ Andersen talked about this recently at Berkeley: math.berkeley.edu/~harold/RTGconference.html There seems to be a new ingredient identifying the different representations: front.math.ucdavis.edu/1110.5027 This paper states that the results are used in the Property (T) paper. But there still seems to be some missing results from the paper (Theorem 5). $\endgroup$
    – Ian Agol
    Commented Feb 2, 2012 at 8:57
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    $\begingroup$ @Ian: What does it mean, "missing results from the paper"? A proof either exists or it does not. Is it correct that there is no proof and the conjecture is still open? $\endgroup$
    – user6976
    Commented Feb 2, 2012 at 10:57
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    $\begingroup$ @Mark: If you look at the paper, you'll see that Theorem 5 cites a theorem in an paper which is "in preparation". Maybe Joergen can address the status of this theorem. $\endgroup$
    – Ian Agol
    Commented Feb 2, 2012 at 16:51
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    $\begingroup$ Meta thread: tea.mathoverflow.net/discussion/1297/… . If there is any more discussion of whether this question should be closed, meta is a better venue. $\endgroup$
    – HJRW
    Commented Feb 2, 2012 at 19:33
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    $\begingroup$ Michael, I hope you have a look at the meta thread. When you vote to close you have to pick one of half-a-dozen fixed "reasons," but I don't think most people who voted to close think this was necessarily a bad question or that you were trying to be argumentative. I don't want you to feel unwelcome. $\endgroup$
    – Noah Snyder
    Commented Feb 3, 2012 at 3:42

1 Answer 1

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Dear Michael,

There certainly is a complete proof of this result including a proof of theorem 5. Theorem 5 follows essentially from my joint work with Kenji Ueno presented in a series of 4 joint papers. These papers are all on the archive and three of them has been published/accepted for publication and the four is submitted for publication. I am currently writing up the paper [A6] "Mapping class group invariant unitarity of the Hitchin connection over Teichmuller space", where a detailed argument for Theorem 5 will be given. It is Corollary 1 of that paper. As soon as that paper is finished I will put it on the archive and I will submit that paper together with my paper "The mapping class group does not have Property T" for publication.

I am not aware of any problems with my proof.- I have ever only received one email suggesting there was a problem (from Mark Sapir), which claimed that Vaughan Jones knew of a problem with the paper. I immediately wrote to Vaughan to ask him what this problem was and he applied right away he didn't know of any mistakes in my argument. Mark right after acknowledged via email that he had misunderstood Vaughan.

I am more than happy to answer all questions via email regarding this result and its proof, so Michael, if you would please let me know who you are and your email address, I will get in touch right away.

Yours,

Jørgen Ellegaard Andersen

Note added by Ian Agol: Andersen's paper proving Theorem 5 appeared on the Arxiv recently.

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    $\begingroup$ Joergen, could you please say what is the status of Theorem 5 to which Ian Agol refers in his comment. $\endgroup$
    – user6976
    Commented Feb 2, 2012 at 17:01
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    $\begingroup$ Thanks for the clarification! It was hard to tell from your initial answer whether you were claiming that the complete proof had already appeared. I fear that this experience will give you a negative impression of MO, but if you poke around a little hopefully you'll see that the focus of the site is on specific mathematical questions, not situations like this one. $\endgroup$
    – Noah Snyder
    Commented Feb 6, 2012 at 17:13

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