Let $S$ be a compact orientable 2-surface with $\chi(X)\leq -3$. Y.Minsky and H.Masur proved that the curve complex of $X$ is $\delta$-hyperbolic and infinite. E.Klarreich and Ursula Hamenstadt(see also U.Hamenstadt) proved the gromovGromov boundary of the curve complex of $S$ is bijective to the collection of minimal filled laminationending laminations. Denote the collection of minimal filled laminationending laminations by $B$.
Note: filledending lamination meansimplies that its complement in $S$ is either a polygon or onecollection of (once-punctured polygon) ideal polygons.
My question is : Given a number $N$, is there possible that there is a collection of essential simpplesimple closed curves $Y=\{c_{\eta}, \eta\in B\}$$Y=\{c_{\eta} \mid \eta\in B\}$ such that $d_{\mathcal {C}(S)}(c_{\eta}, c_{\zeta})\geq N$, for any $\eta, \zeta \in B$ and $\eta\neq \zeta$$\eta \neq \zeta$?