hyperbolic if and only if $G$ and $H$ are synchronized LH.

Maximal elementary subgroup $E(g)$ of a non elementary hyperbolic group $G$, containing an element $g$ of infinite order, is characterized by $E(g)=\{x\in G| \ x^{-1}g^nx=g^{\pm n} \ for \ some \ n=n(x)\in\mathbb{N}-\{0 \}\}$. $E(g)$ in a hyperbolic group. In section \[section4\] of this article we take this algebraic definition (denote by $E^{\mathcal{L}}(g)$) and characterize for lacunary hyperbolic groups.

Let $G$ be a lacunary hyperbolic group and $g\in G$ be an infinite order element. Then $E^{\mathcal{L}}(g)$ has a locally finite normal subgroup $N\triangleleft G$ such that:\
Either $E^{\mathcal{L}}(g)/N$ is an abelian group of Rank 1(i.e. $E^{\mathcal{L}}(g)/N<(\mathbb{Q},+,\cdot )$) or $E^{\mathcal{L}}(g)/N$ is an extension of a rank one group by involutive automorphism(i.e, $a\rightarrow a^{-1}$).

Conversely, for any rank one abelian group $E$, there exists a lacunary hyperbolic group $G$ with an infinite order element $g\in G$ such that $E=E^{\mathcal{L}}(g)$. More generally we obtained the following,

\[rankoneintro\] For any torsion free non elementary hyperbolic group $G$ and a countable family $\mathscr{F}:=\{Q^i_m\}_{i\in\mathbb{N}}$ of subgroups of $(\mathbb Q,+)$, there exists a non elementary, torsion free, non abelian lacunary hyperbolic quotient $G^{\mathscr{F}}$ of $G$ such that the set of all maximal subgroups of $G^{\mathscr{F}}$ is equal 