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

Abstract

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