lacunary hyperbolic group without the unique product property $G^{Q_m}$ such that any maximal subgroup of $G^{Q_m}$ is isomorphic to $Q_m$.

In particular, there exist a non trivial property $(T)$ torsion free divisible lacunary hyperbolic group without the unique product property.

In subsection \[rankonebyfinitesection\] we discuss locally finite by rank one abelian subgroups of lacunary hyperbolic groups. We denote the class of increasing union of elementary subgroups as ${}_{rk-1}\mathscr{E}_{F}$ (see Notation \[notationelementary\]). Even though Theorem \[rankoneintro\]  can be thought of as a corollary of the following theorem, we choose to discuss those in two different subsections in order to make a clear exposition.

Let $G$ be a torsion free non elementary hyperbolic group and $\mathscr{C}:=\{E^j \}_{j\in\mathbb{N}}$ be a countable collection of groups with $E^j\in {}_{rk-1}\mathscr{E}_{F}$ for all $j\geq 1$. Then there exists a non elementary lacunary hyperbolic quotient $G^{\mathscr{C}}$ of $G$ such that $\{E^{{\mathcal{L}}}(h)\ | \ h\in (G^{\mathscr{C}})^0 \}=\mathscr{C}$.

Moreover $\mathscr{C}$ is the set of all maximal proper subgroups of the group $G^{\mathscr{C}}$.

We also investigate Rips type construction in this article in order to construct lacunary hyperbolic groups from a given finitely generated group by closely following the proof strategy of E. Rips for hyperbolic group along with small cancellation 