Teichm\"uller polynomials, Alexander polynomials and finite covers of surfaces

TL;DR

This paper investigates the relationship between finite covers of surfaces, the Teichmüller polynomial, and the action of pseudo-Anosov homeomorphisms on homology, revealing spectral bounds and polynomial governing dynamics.

Contribution

It establishes a connection between the spectrum of surface automorphisms on finite abelian covers and the Teichmüller polynomial, extending understanding of surface dynamics.

Findings

Spectrum of the action is bounded away from the dilatation outside the trivial eigenspace.

Eigenspaces are parametrized by rational points on a torus.

Teichmüller polynomial governs the action on these eigenspaces.

Abstract

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.