The first Cohomology of Affine % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacq % WIKeIOdaahaaWcbeqaaiaadchaaaaaaa!3AFC! $$ \mathbb{Z}^{p} $$ -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 \( \mathbb{R}^{p} \).