First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity

This is the first in a series of papers exploring rigidity properties of hyperbolic actions ofZ ^k orR ^k fork ≥ 2. We show that for all known irreducible examples, the cohomology of smooth cocycles over these actions is trivial. We also obtain similar Hölder and C¹ results via a generalization of the Livshitz theorem for Anosov flows. As a consequence, there are only trivial smooth or Hölder time changes for these actions (up to an automorphism). Furthermore, small perturbations of these actions are Hölder conjugate and preserve a smooth volume.