Rigidity of the Alvarez class

Let $(M,\mathcal{F})$ be a closed manifold with a Riemannian foliation. The \'{A}lvarez class is the cohomology class of degree 1 of $M$ whose triviality characterizes the minimizability of $(M,\mathcal{F})$. We show that the integral of the \'{A}lvarez class along every closed path in $M$ is the logarism of an algebraic integer if $\pi_{1}M$ is polycyclic or $\mathcal{F}$ is of polynomial growth.