Volume 24, issue 4 (2020)

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

Geometry & Topology 24 (2020) 1717–1750
Abstract

Let $\Sigma$ be a compact orientable surface of finite type with at least one boundary component. Let $f\in Mod\left(\Sigma \right)$ be a pseudo-Anosov mapping class. We prove a conjecture of McMullen by showing that there exists a finite cover $\stackrel{̃}{\Sigma }\to \Sigma$ and a lift $\stackrel{̃}{f}$ of $f$ such that ${\stackrel{̃}{f}}_{\ast }:{H}_{1}\left(\stackrel{̃}{\Sigma },ℤ\right)\to {H}_{1}\left(\stackrel{̃}{\Sigma },ℤ\right)$ has an eigenvalue off the unit circle.

Keywords
low-dimensional topology, mapping class groups, representation theory
Mathematical Subject Classification 2010
Primary: 20C12, 57M05, 57M60