condition developed by A. Olshanskii in [@Ol93], later in [@OOS07] by A.O’lshanskii, D. Osin and M. Sapir, and in [@Os'10] by D. Osin.

\[ripsintro\] Let $Q$ be a finitely generated group and $H$ be a non elementary hyperbolic group. Then there exist groups $G$ and $N$, for which we get a short exact sequence; $$1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$$ such that;

1.  $G$ is a lacunary hyperbolic group.

2.  $N$ is a $2$ generated non elementary quotient of $H$.

3.  If $H$ is torsion free then so are $G$ and $N$.

$G$ cannot be hyperbolic if $Q$ has no finite presentation.

Note that if one takes $H$ to be a torsion free, property $(T)$, hyperbolic group without the unique product property in the Theorem \[ripsintro\], then one can recover the following theorem (see, Section 5 for details),

Let $Q$ be a finitely generated group. Then there exists a short exact sequence $$\begin{aligned}
    1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1\nonumber 
    \end{aligned}$$

such that;

-   $G$ is a torsion-free group without the unique product property which is a direct limit of Gromov hyperbolic groups.

-   $N$ is a subgroup of $G$ with Kazhdan’s Property $(T)$ and without the unique product property.

The group von 