The first Cohomology of Affine ℤ[supp] - actions on Tori and applications to rigidity.

Let φ a minimal affine Z[supp]-action on the torus T[supq], p ≥ 2 and q ≥ 1. The cohomology of φ (see definition below) depends on both the algebraic properties of the induced action on H[sup1](T[supq], Z) and the arithmetical properties of the translation cocycle. We give a Diophantine condition that characterizes those affine actions whose first cohomology group is finite dimensional. In this case it is necessarily isomorphic to R[supp]. Thus the R[supp]-action F[subφ] obtained by suspension of φ is parameter rigid, i.e., any other R[supp]-action with the same orbit foliation is smoothly conjugate to a reparametrization of F[subφ] by an automorphism of R[supp]. [ABSTRACT FROM AUTHOR]