Polynomial Invariants for Fibered 3-Manifolds and Teichmuller Geodesics for Foliations

Abstract

Let Image be a fibered face of the Thurston norm ball for a hyperbolic 3-manifold M. Any Image determines a measured foliation Image of M. Generalizing the case of Teichmüller geodesics and fibrations, we show Image 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 Image whose roots determine K(φ). The Newton polygon of ΘF allows one to compute fibered faces in practice, as we illustrate for closed braids in S3. Using fibrations we also obtain a simple proof that the shortest geodesic on moduli space Image has length O(1/g).