Immersed essential surfaces and Dehn surgery

It is known that an embedded essential surface F in a hyperbolic manifold M remains essential in Dehn filling spaces M(γ) for most slopes γ on a torus boundary component T of M. The main theorem of this paper is to generalize this result to immersed surfaces. More explicitly, if an immersed essential surface F has coannular slopes β1,…,βn on T, then there is a constant K such that F remains essential in M(γ) when Δ(γ,βi)>K for all i. It will also be shown that all but finitely many Freedman tubings of a geometrically finite surface in M are π1-injective.