Virtual homological eigenvalue and mapping torus of pseudo-Anosov maps

In this note, we show that, if a pseudo-Anosov map $\varphi :S\to S$ 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 $M_{\varphi }$ has the same property.