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

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.