Subgroups of Genus-2 Quasi-Fuchsian groups and Cocompact Kleinian Groups

In this paper, we want to control the geometry of some surface subgroups of a cocompact Kleinian group. More precisely, provided any genus-2 quasi-Fuchsian group $\mathrm{\Gamma }$ and cocompact Kleinian group $G$, then for any $K>1$, we will find a surface subgroup $H$ of $G$ that is $K$-quasiconformally conjugate to a finite index subgroup $F<\mathrm{\Gamma }$.