Homological eigenvalues of lifts of pseudo-Anosov mapping classes to finite covers

TL;DR

This paper proves McMullen's conjecture by demonstrating that certain lifts of pseudo-Anosov mapping classes to finite covers have eigenvalues off the unit circle in their induced action on homology.

Contribution

It establishes the existence of finite covers and lifts of pseudo-Anosov classes with non-unit eigenvalues on homology, confirming a key conjecture in the field.

Findings

Existence of finite covers with eigenvalues off the unit circle

Lifts of pseudo-Anosov classes can have non-trivial homological eigenvalues

Supports conjecture by McMullen on eigenvalues of lifts

Abstract

Let $\Sigma$ be a compact orientable surface of finite type with at least one boundary component. Let $f \in \textup{Mod}(\Sigma)$ be a pseudo Anosov mapping class. We prove a conjecture of McMullen by showing that there exists a finite cover $\widetilde{\Sigma} \to \Sigma$ and a lift $\widetilde{f}$ of $f$ such that $\wt{f}_*: H_1(\wt{\Sigma}; \mathbb{Z}) \to H_1(\wt{\Sigma}; \mathbb{Z})$ has an eigenvalue off the unit circle.