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$\DeclareMathOperator\Mod{Mod}$Let $\Mod(S)$ be the mapping class group of a closed oriented surface $S$ of genus at least $3$. My question is easy to state: is it currently known whether or not $\Mod(S)$ has Kazhdan's Property (T)? I restrict to surfaces of genus at least $3$ since the mapping class group is trivial in genus 0 (so "yes" for silly reasons), is virtually a nontrivial free group in genus 1 (so "no" in that case), and is known to virtually surject onto the integers in genus 2 (so again "no" in that case).

Whether or not this question is open seems to be a matter of some dispute, for instance on the decade-old MO post Mapping class group and property (T). For those of us who work in adjacent fields, it can be hard to figure out what is going on. Given all the consequences of Property (T) that one might want to use to prove other theorems (e.g., the vanishing of the virtual first Betti number), this is a very frustrating state of affairs.

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    $\begingroup$ @NoahSnyder: I'm not directly asking whether this specific 15 year old preprint is correct or not. In fact, I thought about not mentioning the preprint at all (and would be happy to delete the reference to it), though I expect that it would come up in any discussion. A perfectly good answer would be a pointer to another paper proving or disproving it. More generally, while I think it is totally reasonable to forbid discussion of the correctness of papers, there has to be some kind of statute of limitations. $\endgroup$
    – Thomas
    Aug 4, 2022 at 20:35
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    $\begingroup$ (I mean, at this point this is almost more a matter of the historical record. A substantial number of active MO users weren't even in college when this paper came out. Does it make sense to allow a single unpublished preprint to wall off all discussion of a topic forever?) $\endgroup$
    – Thomas
    Aug 4, 2022 at 20:40
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    $\begingroup$ I will again side with being permissive about this type of question. When it's genuinely unclear what the expert consensus about the status of an important problem in a field is, it's hard to imagine any better venue than MO for sussing that out. $\endgroup$
    – Sam Hopkins
    Aug 4, 2022 at 20:43
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    $\begingroup$ My sense is that the problem is still regarded as open by many people in the field. But it would be great to hear from someone with more direct knowledge. $\endgroup$
    – HJRW
    Aug 5, 2022 at 9:46
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    $\begingroup$ @NoahSnyder: It would be great if an expert would do that! But since they haven't, it doesn't solve the problem for the many people in the field who would like to know the status of this important result. Your comment may be a good account of what "should" happen, but doesn't suggest a useful way forward for the OP. $\endgroup$
    – HJRW
    Aug 5, 2022 at 16:01

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