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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 $M<\infty$.

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Perhaps it would be better if you explain what $\Lambda$ is. What is that constant for the free group or a virtually free group? –  Mark Sapir Mar 7 '12 at 2:11
    
Let say that an approximate identity $e_{\alpha}$ is $C$-completely bounded the completely bounded norm of the net $E_{\alpha}$ is bounded by $C$. The constant $\Lambda_{G}$ for a locally compact group $G$ is defined as $$ \Lambda_{G} = \\{C~|~\exists \textrm{$C$-completely bounded approximate identity in A(G)}\\}.$$ For the free group over 2 generators $F_2$, U. Haagerup proved that $\Lambda_{F_2} = 1$. –  Denis Poulin Mar 7 '12 at 3:14
    
Let say that an approximate identity $e_{\alpha}$ in $A(G)$ is $C$-completely bounded the completely bounded norm of the net $e_{\alpha}$ is bounded by $C$. The constant $\Lambda_{G}$ for a locally compact group $G$ is the infimum of such $C$ over all the $C$-completely bounded approximate identity of $A(G)$. For the free group over 2 generators $F_2$, U. Haagerup proved that $\Lambda_{F_2} = 1$. –  Denis Poulin Mar 7 '12 at 3:16
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Lattices in $Sp(n,1)$ are hyperbolic and have Cowling Haagerup constant $2n-1$. It is precisely from this result that the term "Cowling-Haagerup constant" comes from. –  Mikael de la Salle Mar 7 '12 at 6:52
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Dear all, the main reason of this question is a long standing conjecture which states that the Arens regularity of the Fourier algebra implies that the group is finite. M. Neufang and I proved this for all weakly amenable groups. Now, the only possible counter examples of this conjecture are Tarski's monster groups described with the characterization lacunary hyperbolic groups. However, the general question can be formulated as if $G$ is the direct limits of weakly amenable groups $G_i$ with $\Lambda_{G_i} < M$ for all $i$, is $G$ weakly amenable or at least have the Haagerup property. –  Denis Poulin Mar 7 '12 at 16:08

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