Characteristic subsurfaces and Dehn filling
  Let M be a compact, orientable, irreducible, atoroidal 3-manifold with boundary an incompressible torus. Techniques based on the characteristic submanifold theory are used to bound the intersection number of two slopes \alpha and \beta on the boundary of M. The method applies when \beta is the boundary slope of an essential surface F that is not a semi-fiber (i.e. F is not a fiber and does not split M into two twisted I-bundles), and the Dehn filling M(\alpha) contains a suitable singular surface. One of the main results is that if F is planar and if the fundamental group of M(\alpha) does not contain a non-abelian free subgroup then the intersection number of \alpha and \beta is at most 5. 
Steven Boyer, Marc Culler, Peter B. Shalen, Xingru Zhang