We show C 1 local rigidity for ‫ޚ‬ k .k 2/ higher rank partially hyperbolic actions by toral automorphisms, using a generalization of the KAM (KolmogorovArnold-Moser) iterative scheme. We also prove the existence of irreducible genuinely partially hyperbolic higher rank actions on any torus ‫ޔ‬ N for any even N 6.

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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 generating relations in the group SL n .

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