Teichmueller polynomials, Alexander polynomials and finite covers of surfaces

I will discuss how polynomial invariants for fibered  hyperbolic 3--manifolds can be used
to study homological eigenvalues  of pseudo-Anosov homeomorphisms on finite covers of
surfaces.  In  particular, I will show that if a pseudo-Anosov homeomorphism f has an 
eigenvalue off the unit circle in addition to nontrivial invariant  integral cohomology, then
there is a sequence of finite abelian covers  of the surface such that a definite proportion
(which depends only on  f) of the homological eigenvalues of f on those covers lie off the 
unit circle. It is an open problem to determine whether for an arbitrary  pseudo-Anosov
homeomorphism f there exists a finite cover S' of the  base surface S such that the action
of f on the homology of S' has an  eigenvalue off of the unit circle.  I will show that the
existence of  such a cover is equivalent to the existence of a sequence of finite  covers
of the suspension M_f which have exponential growth of torsion  homology.