# 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. E.Klarreich (see also U.Hamenstadt) proved the Gromov boundary of the curve complex of $S$ is bijective to the collection of ending laminations. Denote the collection of ending laminations by $B$.

Note: ending lamination implies that its complement in $S$ is a collection of (once-punctured) ideal polygons.

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

• What do you mean by "$Y=c_\eta, \eta\in B$"? What is $c_eta$? Do you mean that $c_\eta$ is a sequence of scc's $c_{\eta,n}$ representing the ideal point $\eta$? Then the answer is obviously yes: $$\lim_{n\to\infty} d(c_{\eta,n}, c_{\zeta,n})=\infty,$$ and it holds for any $\delta$-hyperbolic geodesic space, simply by the definition of Gromov boundary. – Misha Nov 4 '12 at 14:06
• Actually, you do not even need the space to be geodesic for this to hold, although the curve complex is geodesic. – Misha Nov 4 '12 at 14:52
• To correct your reference, the theorem equating the Gromov boundary of the curve complex to the space of ending laminations'' is due not to Hamenstadt but to Klarreich www.ericaklarreich.com/page15.html – Lee Mosher Nov 4 '12 at 17:10
• @Misha, $Y$ is defined to be a collection of essential closed curves. I want to write it as a collection, but I don't know how to do it. And the number $N$ depends only on the surface $S$. – yanqing Nov 4 '12 at 23:34
• @Lee Mosher, you are right. I just correct it. – yanqing Nov 4 '12 at 23:37

• Who knows what OP means by $c_\eta$: Could be a single vertex (then, of course, the map is not going to be 1-1), could be a sequence, could be something completely different... – Misha Nov 5 '12 at 23:15
• $c_{\eta}$ is a single vertex. – yanqing Nov 6 '12 at 0:02