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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 finite.

For the definitions of Kazhdan's Property $(T)$ and residually finite 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 finite + Property X $\Rightarrow$ Kazhdan's Property $(T)$.

Maybe there is something similar in the literature?

EDIT: Definition for Property $F$ of the cited paper.

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    $\begingroup$ Maybe you should also state the factorization property; the other two are well known. $\endgroup$ May 22, 2016 at 21:28
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    $\begingroup$ I think this problem would be easier if you required $G$ to be finitely presentable (does Property F imply this?). This is because there is a version of Rips construction, due to Olliver and Wise: for every finitely presented $Q$ there exists a finitely generated (but not finitely presentable), residually finite group $N$ with Property $T$, some (hyperbolic) group $H$, and a short exact sequence $1\rightarrow N\rightarrow H\rightarrow Q\rightarrow 1$. This sequence allows you to prove that $N$ can have certain pathological properties. $\endgroup$
    – user1729
    May 24, 2016 at 10:03
  • $\begingroup$ (An easy property is insoluble membership problem in $H$, by taking $Q$ to have insoluble membership problem. But this is a property of the embedding, not of the group.) $\endgroup$
    – user1729
    May 24, 2016 at 10:03
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    $\begingroup$ I am very skeptical that there is a meaningful Property X. I suggest, however, to ask this question on Mathoverflow. $\endgroup$ May 24, 2016 at 22:51
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    $\begingroup$ @studiosus. I have done so, (mathoverflow.net/questions/239723/…) $\endgroup$
    – M.U.
    May 25, 2016 at 11:46

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