Geometrically Infinite Surfaces in 3–manifolds with Hyperbolic Fundamental Group

We prove a partial generalization of Bonahon’s tameness result to surfaces inside irreducible 3–manifolds with hyperbolic fundamental group. Bonahon’s result states that geometrically infinite ends of freely indecomposable hyperbolic 3–manifolds are simply degenerate. It is easy to see that a geometrically infinite end gives rise to a sequence of curves on the corresponding surface whose geodesic representatives are not contained in any compact set. The main step in his proof is showing that one may assume that these curves are simple on the surface. In this paper, we generalize the main step of Bonahon’s proof, showing that a geometrically infinite end gives rise to a sequence of simple surface curves whose geodesic representatives are not contained in any compact set.