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 $\varphi \in \text{Mod}(\Sigma )$ be a an infinite order mapping class. Then there exists a finite solvable cover $\hat{\Sigma }\to \Sigma$, and a lift $\hat{\varphi }$ of $\varphi$ such that the action of $\hat{\varphi }$ on $H_1(\hat{\Sigma },\mathbb{Z})$ has infinite order. Our main tools are the theory of homological shadows, which was previously developed by the author, and Fourier analysis

Cited by in corpus (4)

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• Virtual homological spectral radii for automorphisms of surfaces
• Prescribed virtual homological torsion of 3-manifolds