Virtual homological eigenvalue and mapping torus of pseudo-Anosov maps

TL;DR

This paper explores the spectral properties of monodromies in fibered 3-manifolds derived from pseudo-Anosov maps, showing that certain homological eigenvalue conditions are preserved under finite covers.

Contribution

It establishes that if a pseudo-Anosov map has a finite cover with spectral radius greater than one on homology, then this property persists in all finite covers of its mapping torus.

Findings

Spectral radius > 1 condition is preserved in finite covers

Monodromies of all finite covers share the spectral property

Connects homological eigenvalues with fibered 3-manifold structures

Abstract

In this note, we show that, if a pseudo-Anosov map $\phi:S\to S$ admits a finite cover whose action on the first homology has spectral radius greater than $1$, then the monodromy of any fibered structure of any finite cover of the mapping torus $M_{\phi}$ has the same property.