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I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I may ask it here too.

There is a theorem that states that a discrete group $G$ with Kazhdan's Property $(T)$ and Property $F$ (so called factorisation property) is residually finite (see Kirchberg, Discrete groups with Kazhdan' s property T and factorization property are residually finite), i.e.

Kazhdan's Property $(T)$ + Property F $\Rightarrow$ Residual finiteness.

For the definitions of Kazhdan's Property $(T)$ and residual finiteness see e.g. the corresponding wiki-articles.

I am wondering if some kind of "converse" is true. More precisely, I am looking for some property, let us call it Property X, such that:

Residual finiteness + Property X $\Rightarrow$ Kazhdan's Property $(T)$.

Maybe there is something similar in the literature?

Definition for Property $F$ of the cited paper.

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    $\begingroup$ One says "residual finiteness" but "residually finite". $\endgroup$ – YCor May 25 '16 at 13:06
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    $\begingroup$ I don't expect any interesting answer other than tautologies such as: Property X = Property T, or Property X= (Property T or not residually finite), or some kind of very restrictive Property. Maybe you have some more specific question in mind (what kind of property?) $\endgroup$ – YCor May 25 '16 at 13:09
  • $\begingroup$ @YCor. What are your arguments for that expectation? $\endgroup$ – M.U. May 26 '16 at 17:33
  • $\begingroup$ I don't claim any argument, it's just my intuition, and also my natural reaction to such a vague question. (Well my claim is possibly in contradiction with Agol's answer but I'm unable to say if the "property X" he refers to is very strong restrictive condition, e.g., whether it implies property T for $SL_3(\mathbf{Z})$ or some infinite hyperbolic groups). $\endgroup$ – YCor May 26 '16 at 21:25
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Rufus Willett and Guoliang Yu, MR 3246936 Geometric property (T), Chin. Ann. Math. Ser. B 35 (2014), no. 5, 761--800. showed that if a finitely generated group is residually finite and finite quotients of the Cayley graph have ``Geometric Property (T)", then the group has property (T). A residually finite group has property $(\tau)$ if finite quotients of the Cayley graph are expanders, and it is known that there are groups which possess property $(\tau)$ but not property (T). So this condition on a sequence of finite graphs is stronger than just being expanders, and it is a property holding under a very coarse equivalence relation.

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Perhaps it's worth mentioning that Property (T) seems to repel certain strengthenings of residual finiteness. An open question of Long and Reid asks:

Is there an infinite finitely generated group which is LERF and has Property (T)?

Recall that LERF stands for Locally Extended Residually Finite, and means that every finitely generated subgroup is closed in the profinite topology. (Residual finiteness means the trivial subgroup is closed)

Long and Reid's question may have a positive answer, but the point of it is that such groups seem in any case to be extremely rare and/or difficult to construct.

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