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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 $T^2$.

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up vote 6 down vote accepted

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:

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Thanks Frohman. So it seems all ? of the subtle part of my question. I even don't know whether there exists a three manifold without P-A surface. PS, for Anosov case, the second ? of the subtle part is a question in the paper which motivate my question. – Bin Yu Nov 8 '11 at 2:43
I don't know what the answer to the first part of your question is. I think its more subtle than the second part. – Charlie Frohman Nov 8 '11 at 14:30
Sorry, "So it seems all ? of the subtle part of my question" should be "So it seems the answers of all ? of the subtle part of my question are negative" – Bin Yu Nov 9 '11 at 7:15

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