$q:\mathbb{R}\rightarrow (X,d_X)$ such that there exists $\lambda,c>0$ so that for all $s,t\in \mathbb{R}$, $$\frac{|s-t|}{\lambda}-c < d_X(f(x),f(y))< \lambda|s-t|+c$$ is called a $(\lambda,c)$ quasi geodesic.

[@OOS07 Theorem 3.18 ]\[aa\] Let $G$ be a lacunary hyperbolic group. Then, every finitely generated undistorted (i.e, quasi-isometrically embedded) subgroup of $G$ is lacunary hyperbolic.

So it is natural to expect the “same" result to be true for lacunary hyperbolic space.

\[qisubspace\] Let $X$ be a lacunary hyperbolic space and $q: Y\rightarrow X$ is a quasi isometric embedding. Then $Y$ is also lacunary hyperbolic space.

Proof follows from similar argument from Theorem ([\[aa\]]{}).

Suppose $X,Y$ are two quasi-isometric geodesic metric space. Then $X$ is lacunary hyperbolic if and only if $Y$ is lacunary hyperbolic.

\[synchrodef\] Let $G_i$ be LHG (lacunary hyperbolic group) for $i=1,2,\cdots ,k$ with corresponding hyperbolicity function $f_{G_i}$ for $G_i$ for all $i=1,2,\cdots,k$. Then we call $G_i$’s are synchronized LH if there exists an increasing infinite sequence $\{x_j\}_{j\in \mathbb{N}}$ of real numbers and finite generating sets $X_i$ of $G_i$ for all $i$ such that the hyperbolicity function with respect to generating sets $X_i$, $f_{G_i}$ satisfies $\lim_{j\rightarrow\infty}\frac{f_{G_i}(x_j)}{x_j}=0$ for all $i=1,2,\cdots,k$ (equivalently $\liminf_{t\rightarrow \infty} \frac{\sum_{i=1}^{k}f_{G_i}(t)}{t}=0$).

Free product of two lacunary hyperbolic group is not necessarily a lacunary 