Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper "non-stationary" generalization of the classical theory of normal forms for local contractions.Comment: 28 pages, LaTe

Local Rigidity of Partially Hyperbolic Actions: Solution of the General Problem via KAM Method

We consider a broad class of partially hyperbolic algebraic actions of higher-rank abelian groups. Those actions appear as restrictions of full Cartan actions on homogeneous spaces of Lie groups and their factors by compact subgroups of the centralizer. The common property of those actions is that hyperbolic directions generate the whole tangent space. For these actions we prove differentiable rigidity for perturbations of sufficiently high regularity. The method of proof is KAM type iteration scheme. The principal difference with previous work that used similar methods is very general nature of our proofs: the only tool from analysis on groups is exponential decay of matrix coefficients and no specific information about unitary representations is required