Let $S$ be a closed incompressible surface in a finite volume hyperbolic 3-manifold $M$ without cusps. Let $N$ be the cover of $M$ associated to $\pi\_1(S) \subset \pi\_1(M)$. The cover $N$ is homeomorphic to $S\times\mathbb{R}$ and has two ends. An end is *convex cocompact* if it has a neighborhood whose intersection with the convex core of $N$ is bounded and *degenerate* otherwise.


If $S$ is a fiber, then both ends of $N$ are degenerate. If the group of isometries corresponding to $\pi\_1(S)$ is quasi-Fuchsian, then both ends of $N$ are convex cocompact.


My questions: Is it possible for one end of $N$ to be convex cocompact and the other to be degenerate?  

What if we allow $M$ to have infinite volume?
Can you point me to a description of an example or an explanation of why there are no examples? 



Hyperbolic structures on $S\times\mathbb{R}$
It follows from Thurston's Covering Theorem that there are no such examples.


The covering theorem says that if a degenerate end is infinite-to-one under a covering map, then you are (virtually) in the fibered case. See R.D. Canary, A covering theorem for hyperbolic 3-manifolds and its applications, Topology 35:3 (1996), 751–778.


