Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations

Abstract

Let F⊂H 1(M 3, R) be a fibered face of the Thurston norm ball for a hyperbolic 3-manifold M. Any φ∈ R +·F determines a measured foliation F of M. Generalizing the case of Teichmüller geodesics and fibrations, we show F carries a canonical Riemann surface structure on its leaves, and a transverse Teichmüller flow with pseudo-Anosov expansion factor K( φ)>1. We introduce a polynomial invariant Θ F∈ Z[H 1(M, Z)/ torsion] whose roots determine K( φ). The Newton polygon of Θ F allows one to compute fibered faces in practice, as we illustrate for closed braids in S 3. Using fibrations we also obtain a simple proof that the shortest geodesic on moduli space M g has length O(1/ g).