The first Cohomology of Affine $\mathbb{Z}^{p}$-actions on Tori and applications to rigidity*

Abstract.

Let ϕ a minimal affine $\mathbb{Z}^{p}$-action on the torus T ^q, p ≥ 2 and q ≥ 1. The cohomology of ϕ (see definition below) depends on both the algebraic properties of the induced action on H ¹(T ^q ,$\mathbb{Z}$) 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 $\mathbb{R}^{p}$. Thus the $\mathbb{R}^{p}$-action F _ϕ obtained by suspension of ϕ is parameter rigid, i.e., any other $\mathbb{R}^{p}$-action with the same orbit foliation is smoothly conjugate to a reparametrization of F _ϕ by an automorphism of $\mathbb{R}^{p}$.