Rigidity of the \'Alvarez classes of Riemannian foliations with nilpotent structure Lie algebras

We show that if the structure algebra of a Riemannian foliation F on a closed manifold M is nilpotent, then the integral of the \'Alvarez class of (M,F) along every closed path is the exponential of an algebraic number. By this result and the continuity of the \'Alvarez class under deformations shown in arXiv:1009.1098v2, we prove that the \'Alvarez class and the geometrically tautness of Riemannian foliations on a closed manifold M are invariant under deformation, if the fundamental group of M has polynomial growth.