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On page 13 of the monograph of Bekka, de la Harpe, and Valette on Kazhdan's property (T), it is written "for $n \geq 3$, the compact group $\mathrm{SO}(n)$ has the strong property (T)," citing articles of Shalom and Bekka from 1999 and 2003, respectively. Does this indeed refer to Lafforgue's strong property (T), or to a distinct concept that I am unaware of? I ask because Lafforgue's introduction of strong property (T) came in his paper "Un renforcement de la propriété (T)," which was published in 2008 (that is, later than the aforementioned papers of Shalom and Bekka).

Thank you!

Edit: Thanks to YCor for clarifying in the comments that the "strong property (T)" referred to in my original question is not the same as Lafforgue's strong property (T).

Is it known whether for $n \geq 3$, the group $\mathrm{SO}(n)$ has Lafforgue's strong property (T)?

I may as well also ask the following: does Lafforgue's (T) share any of the same general properties as Kazhdan's (T)? For instance, do compact groups possess Lafforgue's (T)? Is it preserved under taking quotients?

I was unable to find any answers to these questions elsewhere.

Thanks!

Second Edit:

Indeed, compact groups have Lafforgue's Strong Property (T). Mikael de la Salle has some very nice lectures on Strong Property (T), available at this link: http://www.ipam.ucla.edu/abstract/?tid=14627&pcode=ZIM2018

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    $\begingroup$ No, it's unrelated. They refer to another notion of "strong Property T". Namely, a locally compact group has strong Property T (in the original sense) if it admits a finite Kazhdan subset (while ordinary Property T means that it admits a compact Kazhdan subset — in particular for discrete groups the notions are the same). $\endgroup$
    – YCor
    Commented Apr 9, 2023 at 0:32
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    $\begingroup$ I should add that Lafforgue named his property "propriété T renforcée", which was unfaithfully translated into English as "strong Property T" while the correct translation should have been "strengthened Property T". $\endgroup$
    – YCor
    Commented Apr 9, 2023 at 0:34
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    $\begingroup$ In addition, the fact that $\mathrm{SO}(n)$ has strong Property T (in the sense of Shalom 1999 Trans. AMS) directly follows, if I'm correct, from results of the 80s, namely Margulis for $n\ge 5$ and Drinfeld for the even harder case $n=3,4$. Only later the terminology was introduced. $\endgroup$
    – YCor
    Commented Apr 9, 2023 at 0:36
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    $\begingroup$ @YCor thank you very much for the clarification! Do you know whether SO(n) has the "strengthened" (just to be clear!) property (T) of Lafforgue? $\endgroup$
    – Aleksander Skenderi
    Commented Apr 9, 2023 at 0:39
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    $\begingroup$ I'd tend to believe that all compact groups have Lafforgue's strong Property T, but it's better if confirmed by a specialist. $\endgroup$
    – YCor
    Commented Apr 9, 2023 at 0:45

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