Every Infinite order mapping class has an infinite order action on the homology of some finite cover

Abstract

We prove the following well known conjecture: let $\Sigma$ be an oriented surface of finite type whose fundamental group is a nonabelian free group. Let $\phi \in \textup{Mod}(\Sigma)$ be a an infinite order mapping class. Then there exists a finite solvable cover $\widehat{\Sigma} \to \Sigma$, and a lift $\widehat{\phi}$ of $\phi$ such that the action of $\widehat{\phi}$ on $H_1(\widehat{\Sigma}, \mathbb{Z})$ has infinite order. Our main tools are the theory of homological shadows, which was previously developed by the author, and Fourier analysis