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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 gromov Gromov boundary of the curve complex of $S$ is bijective to the collection of minimal filled laminationending laminations. Denote the collection of minimal filled lamination ending laminations by $B$.

Note: filled ending lamination means implies that its complement in $S$ is either a polygon or one-punctured polygoncollection of (once-punctured) ideal polygons.

My question is: Given a number $N$, is there possible that there is a collection of essential simpple simple closed curves $Y={c_{\eta}, Y=\{c_{\eta} \mid \eta\in B}$ B\}$ such that $d_{\mathcal {C}(S)}(c_{\eta}, c_{\zeta})\geq N$, for any $\eta, \zeta \in B$ and $\eta\neq \eta \neq \zeta$?
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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 proved the gromov boundary of the curve complex of $S$ is bijective to the collection of minimal filled lamination. Denote the collection of minimal filled lamination $B$.

Note: filled lamination means its complement in $S$ is either a polygon or one-punctured polygon.

My question is : Given a number $N$, is there possible that there is a collection of essential simpple closed curves $Y={c_{\eta}, \eta\in B}$ such that $d_{\mathcal {C}(S)}(c_{\eta}, c_{\zeta})\geq N$, for any $\eta, \zeta \in B$ and $\eta\neq \zeta$?
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faraway curves in surface

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. Ursula Hamenstadt proved the gromov boundary of the curve complex of $S$ is bijective to the collection of minimal filled lamination. Denote the collection of minimal filled lamination $B$.

Note: filled lamination means its complement in $S$ is either a polygon or one-punctured polygon.

My question is : Given a number $N$, is there possible that there is a collection of essential simpple closed curves $Y={c_{\eta}, \eta\in B}$ such that $d_{\mathcal {C}(S)}(c_{\eta}, c_{\zeta})\geq N$, for any $\eta, \zeta \in B$ and $\eta\neq \zeta$?