Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic ℝ^k actions

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.