$\lim_{i\rightarrow\infty }\frac{f_G(y_i)}{y_i}=0$. So we have $\lim_{i\rightarrow\infty }\frac{f_{G_j}(y_i)}{y_i}=0$ for all $j=1,2$. Hence by definition $G_1,G_2$ are synchronized LH.

\[freeprodlacuremark\] Above proposition is in fact true for free product of finite number of LHGs i.e, for a finite number of lacunary hyperbolic groups $\{G_i\}_{i=1}^{k}$, $G=*^{k}_{i=1}G_i$ is LHG if and only if $\{G_i\}$ are synchronized LH. In particular $*_{i=1}^kG$ is a lacunary hyperbolic group whenever $G$ is a lacunary hyperbolic group for any positive integer $k$.

Algebraic properties of lacunary hyperbolic groups {#section4}
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Small cancellation theory
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### van Kampen diagrams

Given a word $W$ in alphabets $\mathcal{S}$, we denote its length by $\|W\|$. We also write $W\equiv V$ to express the letter-for-letter equality for words $U,V$.

Let $G$ be a group generated by alphabets $\mathcal{S}$. A van Kampen diagram $\triangle$ over a presentation $$\begin{aligned}
 \label{vankampen}
G=\langle S|\mathcal{R}\rangle\end{aligned}$$ where $\mathcal{R}$ is cyclically reduced words over alphabets $S$, is a finite, oriented, connected, planar 2-complex endowed with a labeling function $Lab:E(\triangle)\rightarrow \mathcal{S}^{\pm 1}$, where $E(\triangle)$ denotes the set of oriented edges of $\triangle$, such that $Lab(e^{-1})\equiv (Lab(e))^{-1}$. Given a cell $\Pi$ of $\triangle$, $\partial \Pi$ denotes its boundary; similarly $\partial \triangle$ denote boundary of $\triangle$. The labels of $\partial \triangle$ and $\partial \Pi$ 