Surface Subgroups of Cocompact Kleinian Groups

Kahn and Markovic proved the Surface Subgroup conjecture for closed hyperbolic 3-manifolds more than ten years ago. The surface subgroup they constructed can be as close as possible to Fuchsian. However, a closed hyperbolic 3-manifold can also have surface subgroups far away from being Fuchsian. Our result states that provided any genus-2 quasi-Fuchsian group Γ and cocompact Kleinian group G, then for any K>1, one can find a surface subgroup H of G that is K-quasiconformally conjugate to a finite index subgroup F<Γ. We will point out the difference between the above theorem and the original Surface Subgroup Theorem, discuss the proof idea, and introduce some applications.