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.