| Periodic cycle functionals and Cocycle rigidity for certain partially hyperbolic R k actions (2009) | |||||||||||||||
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| Abstract. We give a proof of cocycle rigidity in Hölder and smooth categories for Cartan actions on SL(n, R)/Γ and SL(n, C)/Γ for n ≥ 3 and Γ cocompact lattice, and for restrictions of those actions to subspaces which contain a two-dimensional plane in general position. This proof does not use harmonic analysis, it relies completely on the structure of stable and unstable foliations of the action. The key new ingredient is the use of the description of relations in the group SLn. 1. Cocycles and cocycle rigidity 1.1. Definitions. Let α: R k ×M → M be an action of R k on a compact Riemannian manifold M. If H is a topological group then a cocycle (or an one-cocycle) over the action α with values in H is a continuous function β: R k × M → H satisfying: (1.1) β(a + b, x) = β(a, α(b, x))β(b, x) for any a, b ∈ R k. A cocycle is cohomologous to a constant cocycle (cocycle not depending on x) if there exists a homomorphism π: R k → H and a continuous transfer map h: M → H such that for all a ∈ R k | |||||||||||||||
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