Homological eigenvalues of mapping classes and torsion homology growth for fibered 3--manifolds

Abstract

Let S be an orientable surface with negative Euler characteristic, let ψ\in\Mod(S) be a mapping class of S, and let T_ψ be the mapping torus of ψ. We study the action of lifts of ψon the homology of finite covers of S via the torsion homology growth of towers of finite covers of T_ψ. We show that ψadmits a lift to a finite cover with a homological eigenvalue of length greater than one if and only if the mapping torus T_ψ admits a finite cover X and a certain tower of abelian covers which have exponential torsion homology growth. We show that the existence of such a lift of ψis intrinsic to T_ψ, in the sense that it does not depend on the particular fibration used to present T_ψ.

This article is withdrawn due to an error in the proof of the main theorem. The result is true and has been established by several other authors

References in corpus (3)

• Exponential growth of torsion in Abelian coverings
• Small dilatation pseudo-Anosovs and 3-manifolds
• Representations of Galois Groups on the Homology of Surfaces