Surface groups in some surgered manifolds

Abstract

We show that closed π 1-injective quasi-Fuchsian surfaces, immersed in a complete hyperbolic 3-manifold of finite volume, will remain π 1-injective after all but finitely many Dehn Surgeries. We use the theory of arithmetic manifolds to construct infinite families of totally geodesic surfaces in the figure-eight knot complement and the Whitehead Link complement. We use these results to show that all surgeries, except 1/0, on the figure-eight knot complement yield manifolds which contain a surface group. Furthermore, we show that all k-twist knots ( k>10) contain a closed, π 1-injective surface which will remain π 1-injective after all but at most 60 surgeries.