Teichm\"uller polynomials, Alexander polynomials and finite covers of surfaces

In this note we explore a connection between finite covers of surfaces and the Teichm"uller polynomial of a fibered face of a hyperbolic 3–manifold. We consider the action of a homological pseudo-Anosov homeomorphism $\psi$ on the homology groups of a class of finite abelian covers of a surface ${\mathrm{\Sigma }}_{g,n}$. 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 $\psi$ on these eigenspaces is bounded away from the dilatation of $\psi$. We show that the action $\psi$ on these eigenspaces is governed by the Teichm"uller polynomial.