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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.

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