Teichmüller polynomials, Alexander polynomials and finite covers of surfaces

In this note 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 $Σ_{g,n}$. 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 $ψ$ 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.

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