Question

A surface $S$ in a three manifold $M$ is pseudo-Anosov means if there exists a homeomorphism $f$ over $M$ for which $S$ is $f$ invariant and $f$ is a pseudo-Anosov on $S$. For example, $M$---- any surface bundle over circle with pseudo-Anosov monodromy map; $S$---- a fiber (surface).

Question: Which three manifolds admit a pseudo-Anosov surface?

More subtle, if $M$ is irreducible, does the example(s) above contain all cases? Moreover, is the following true?: $M$ admits a pseudo-Anosov surface iff there exists a prime factor of $M$ admits a pseudo-Anosov surface?

This question is motivated by link text. In this paper, the authors tell us: if $S$ is torus, $f$ is Anosov and $M$ is irreducible, $M$ must be one of the following 3 cases: (1) the 3-torus $T^3$; (2) the mapping torus of -id; (3) the mapping tori of hyperbolic automorphisms of $T^2$.

Answer 1

For genus greater than one, there are lots of Pseudo-Anosov mapping classes that extend over handlebodies, so you can build lots of examples that are not of the types listed above. Here is a specific example: D.D. Long "Pseudo-Anosov maps which extend over two handlebodies" Proceedings of Edinburgh Society(1990) 33, 181-190

There is recent work of Biringer, Johnson, and Minsky characterizing when a power of a pseudo-anosov extends over a handlebody: http://arxiv.org/abs/1011.0021