Let $M$ be a closed, oriented, hyperbolic $3$-manifold which is a surface bundle over $\mathbb{S}^1$. 


Is there some $\pi\_1$-injective closed surface (perhaps not embedded) $S \subset M$ which is not a fiber, and which does not intersect all the fibers?



Immersed incompressible surfaces in surface bundles
There is no such surface. If $S$ is a surface that misses a fiber then $S$ lies in a submanifold of $M$ that is homeomorphic to a product $F \times I$, where $F$ is the fiber. Any $\pi\_1$-injective closed surface in a product is homotopic to a cover of the fiber $F$. This was shown by Waldhausen for embedded surfaces. The immersed case follows by passing to a covering space.


