Quasi-Fuchsian surface subgroups of infinite covolume Kleinian groups
Given a complete hyperbolic 3-manifold $N$, one can ask whether its fundamental
group $\Gamma=\pi_1N$ contains any quasi-Fuchsian surface subgroups.
Equivalently, given a pared 3-manifold $(M,P)$, one can ask whether there
exists a closed immersed $\pi_1$-injective surface in $M$ that avoids the
peripheral subgroups associated to $P$. This is known to be true for closed
hyperbolic 3-manifolds, and more generally for finite volume hyperbolic
3-manifolds. We outline a strategy to solve the case of infinite volume
hyperbolic 3-manifolds, that is, infinite covolume Kleinian groups. As a first
step in this program, we give a characterization of books of $I$-bundles which
contain quasi-Fuchsian surface subgroups.