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.