Teichmüller polynomials, Alexander polynomials and finite covers of surfaces

Abstract

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 $\psi$ 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 Teichmüller polynomial.

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

References in corpus (2)

• Representations of Galois Groups on the Homology of Surfaces
• The action of mapping classes on nilpotent covers of surfaces

Cited by in corpus (1)

• On the surjectivity of the Symplectic representation of the mapping class group