Abstract: A surface S in a manifold M is filling if S cuts M into contractible components. We prove for any closed hyperbolic 3-manifold M , there exists a K”> 0 such that every homotopy class of K-quasi-Fuchsian surfaces with 1<K ≤ K” is filling. As a corollary, the set of embedded surfaces in M satisfies a dichotomy: it consists of at most finitely many totally geodesic surfaces and surfaces with a quasi-Fuchsian constant lower bound K”. Each of these nearly geodesic surfaces separates any pair of distinct points at the sphere of infinity. Crucial tools include the rigidity results of Mozes-Shah, Ratner, and Shah. This work is inspired by a question of Wu and Xue whether random geodesics on random hyperbolic surfaces are filling.
