Abstract: Kahn and Markovic proved the Surface Subgroup conjecture for compact hyperbolic 3-manifolds more than ten years ago. And the surface subgroup they constructed can be as close as possible to Fuchsian. But a compact hyperbolic 3-manifold can also have surface subgroups far away from being Fuchsian. Actually, provided any genus-2 quasi-Fuchsian group Γ and cocompact Kleinian group G, we can show that 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. As an application of this result, we can prove that the set of Hausdorff dimensions of limit sets of surface subgroups of G is dense in [1,2].
