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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 this decade-old MO post. 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.

$\DeclareMathOperator\Mod{Mod}$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.

$\DeclareMathOperator\Mod{Mod}$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 an infinite 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 this decade-old MO post. 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.

$\DeclareMathOperator\Mod{Mod}$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 this decade-old MO post. 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.

$\DeclareMathOperator\Mod{Mod}$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 an infinite 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).

There is a preprint on the arXiv claiming that it does not, and there is a decade-old MathOverflow question about this paper here in which the author defends it and gives pointers to preprints that fill in some of the details. But as far as I can tell, nothing has been published. This is a very frustrating state of affairs for those of us that work in adjacent fields, so I was hoping that experts could chime in and sort out for the rest of us what is going on.

Of course, one can "extrapolate from silence" that something must be the matter, but that feels very unsatisfying.

$\DeclareMathOperator\Mod{Mod}$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 an infinite 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 this decade-old MO post. 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.