Teichmüller Polynomials, Alexander Polynomials and Finite Covers of Surfaces

In this article we explore a connection between finite covers of surfaces and the Teichmüller polynomial of a fibered face of a hyperbolic 3–manifold. We consider the action of a homological pseudo-Anosov homeomorphism ψ on the homology groups of a class of finite abelian covers of a surface Σg,n. Eigenspaces of the deck group actions on these covers are naturally parametrized by rational points on a torus. We show that away from the trivial eigenspace, the spectrum of the action of ψ on these eigenspaces is bounded away from the dilatation of ψ. We show that the action ψ on these eigenspaces is governed by the Teichmüller polynomial.