Non-abelian cohomology of abelian Anosov actions

We develop a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups (Z(k) and R-k, k greater t han or equal to 2) with values in a linear Lie group. In this paper we cons ider the discrete-time case. Our results apply to cocycles of different reg ularity, from Holder to smooth and real-analytic. The main conclusion is th at the corresponding cohomology trivializes, i.e. that any cocycle from a g iven class is cohomologous to a constant cocycle. The principal novel featu re of our method is its geometric character; no global information about th e action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and t o rigidity problems.