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?