In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's monster group, as the famous example of a non-amenable group containing no free subgroup of rank 2.

In this paper, they characterize lacunary hyperbolic group as groups which are obtain by a direct limits of hyperbolic groups with specific condition on the constant of hyperbolicity.

N. Ozawa proved also that hyperbolic groups are weakly amenable. Weakly amenable group are locally compact groups such that the Fourier algebra over these groups has an approximate identity bounded in the completely bounded multiplier norm. However, we do not know if every hyperbolic groups are $M$-weakly amenable, i.e. there is a constant $M < \infty$ such that the Cowling-Haagerup constant $\Lambda$ is bounded by $M$ for any hyperbolic group. For some subclasses of hyperbolic groups, this is known.

**Question**: What can we say about the lacunary hyperbolic groups which are the direct limits of hyperbolic group $G_i$ such that $\Lambda_{G_i} < M$, for all $i$ and some $M<\infty$.