Virtually Haken Dehn--lling

We show that \most" Dehn-llings of a non-bered, atoroidal, Haken three-manifold with torus boundary are virtually Haken. 1 Results Suppose that X is a compact, oriented, three-manifold with boundary a torus T: We will pick a basis of H 1 (T) represented by simple loops ; such that = 0 in H 1 (X; Q): We call a longitude and a meridian. A slope, ; on T is the isotopy class of essential unoriented simple closed curve. The manifold X() is the result of Dehn-lling along the slope : This means that a solid torus is glued along its boundary to T so that a meridian disc of the solid torus is glued onto : The manifold X is atoroidal if every ZZ subgroup of 1 X is conjugate into 1 T: The distance between two slopes ; is (;) which is the absolute value of the algebraic intersection number of the homology classes represented by these slopes. Theorem 1.1 Suppose that X is a compact, connected, oriented, irreducible, atoroidal three-manifold with boundary a torus T: Suppose that S is a compact, connected, oriented, non-separating, incom-pressible surface properly embedded in X with non-empty boundary. Suppose that S is not a ber of a bration of X over the circle. Let g be the genus of S and b the number of boundary components of S: Then there is an integer 1 P(X; S) 3g ? 2 + b with the following property.