Teichmüller polynomials, Alexander polynomials and finite covers of surfaces

Details Teichmüller polynomials, Alexander polynomials and finite covers of surfaces Teichmüller polynomials, Alexander polynomials and finite covers of surfaces

2011-10-17

arxiv.org/abs/1110.3746v2

Mathematics

Geometric Topology

8 pages, submitted version. Fixed some inaccurate statements in the previous version

Scientific paper

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.

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