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

TL;DR

This paper investigates the relationship between lifts of mapping classes on surfaces and the exponential growth of torsion homology in finite covers of their mapping tori, revealing intrinsic properties of the 3-manifolds involved.

Contribution

It establishes a criterion linking homological eigenvalues of lifts of mapping classes to exponential torsion homology growth in finite covers of fibered 3-manifolds, independent of fibration choices.

Findings

Existence of a lift with eigenvalue > 1 iff exponential torsion homology growth occurs

Homological eigenvalues are intrinsic to the mapping torus, not dependent on specific fibrations

Provides a criterion connecting surface mapping classes to 3-manifold topology

Abstract

Let S be an orientable surface with negative Euler characteristic, let \psi\in\Mod(S) be a mapping class of S, and let T_{\psi} be the mapping torus of \psi. We study the action of lifts of \psi on the homology of finite covers of S via the torsion homology growth of towers of finite covers of T_{\psi}. We show that \psi admits a lift to a finite cover with a homological eigenvalue of length greater than one if and only if the mapping torus T_{\psi} 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 \psi is intrinsic to T_{\psi}, in the sense that it does not depend on the particular fibration used to present T_{\psi}.