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Non-abelian cohomology of abelian Anosov actions. (English) Zbl 0977.57042
The articles of A. N. Livshits [Math. Notes 10, 758-763 (1971); translation from Mat. Zametki 10, 555-564 (1971; Zbl 0227.58006); and Math. USSR, Izv. 6, 1278-1301 (1972); translation from Izv. Akad. Nauk SSSR., Ser. Mat. 36, 1296-1320 (1972; Zbl 0252.58007)] started the study of cocycles over Anosov diffeomorphisms (i.e., actions of the group $$\mathbb{Z}$$ of integers) and flows (i.e., actions of the group $$\mathbb{Z}$$ of real numbers). An important conjecture asserts that tori, nilmanifolds, and infranilmanifolds are the only ones supporting Anosov diffeomorphisms. The articles of Livshits have created a lot of research and the article under review includes a survey of related results and then develops a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups $$\mathbb{Z}^k$$ and $$\mathbb{R}^k$$ $$(k\geq 2)$$ with values in a linear Lie group. The main result of the article asserts that the corresponding cohomology trivializes. The methods applied to get the result have geometric nature and are not using global information about the action based on harmonic analysis. As the authors note, the methods can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.

MSC:
 57S25 Groups acting on specific manifolds 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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