the authors have considered lacunary hyperbolic groups (see definition, sec. \[lacunarydef\])  and showed that there exists a finitely generated group, one of whose asymptotic cone $\mathcal{C}$ has non trivial countable fundamental group ($\pi(\mathcal{C})\equiv \mathbb{Z}$). We studied those limit groups in this article via hyperbolicity function (see section 2,3) and characterized lacunary hyperblic groups by hyperbolicity functions which measures the thinness of triangles in a geodesic metric spaces. We show that the thinness of triangles can grow sublinearly with respect to the size of perimeter of triangles in hyperbolic metric space and thinness of triangle grows sublinearly with respect to an increasing sequence of the size of perimeter of triangles in lacunary hyperbolic metric space.

Let G be a finitely generated group with generating set $S$. Then $G$ is lacunary hyperbolic if and only if the hyperbolicity function $f_{G}$ of the corresponding Cayley graph $\Gamma(G,S)$ satisfies $\liminf_{t\rightarrow\infty}\frac{f_{G}(t)}{t}=0$.

One advantage of having a dynamical characterization is that one can get a necessary and sufficient condition when lacunary hyperbolicity is preserved under free product. In general lacunary hyperbolicity is not stable under free product [@OOS07 Example 3.16]. See Definition \[synchrodef\] for the definition of synchronized LH.

Let $G_1=G* H$. Then $G_1$ is lacunary 