# Extensions of Fried's Theorem: Surface bundles over Circles and Flows on 3 Manifolds

The Thurston Polytope provides a way to organize information about the embedded surfaces living in a 3-manifold.

His amazing theorem, often called the "Fibered Faces" Theorem, says that if you have a homology class of surface $[S]$ representing the fiber of a fibration for $M$ over $S^1$, then not only does $[S]$ lie in the interior of the cone of a top-dimensional face $\sigma$, but every surface in int(cone($\sigma$)) also represents a fiber surface for $M$.

Fried later proved that given a fibered face $\sigma$ of the Thurston Polytope, you can associate to it a flow with some nice properties, namely that any primitive integral class in the cone of $\sigma$ is represented by a cross-section to the flow, and the first return map is pseudo-Anosov. (I believe this theorem can be found in Expose 14 in FLP: "Fibrations of S^1 with Pseudo Anosov Monodromy").

This result was later used by Mosher to associate a single branched surface $B$ to a fibered face $\sigma$, such that $B$ carries fiber representatives of every class in int(cone($\sigma$)). (See "Surfaces and Branched Surfaces Transverse to Pseudo-Anosov Flows on 3-Manifolds" for more details)

My Question: Are there analogous results - either to Fried's theorem or Mosher's - if the manifold has boundary? Namely, what if I'm looking at a link exterior in $S^3$, where I know I have a fibered face? Can I also conclude that such a flow or branched surface exists? I am hopeful that such statements exist, but I haven't find any just yet....

• I realize you are interested in the case of $\chi=0$, but there is also a generalization to sutured manifolds: msp.org/gtm/1999/02/p003.xhtml – Ian Agol Sep 24 '17 at 18:58

## 1 Answer

The theory seems to be quite well-developed in the link complement case. For examples seem Curt McMullen's excellently written (I am being redundant) paper

McMullen, Curtis T., Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations, Ann. Sci. Éc. Norm. Supér. (4) 33, No. 4, 519-560 (2000). ZBL1013.57010.

Or Danny Calegari's blog post. Both have extensive references.