  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
$\Sigma_{g,n}$. 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.

Geometric Topology