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

TL;DR

This paper proves that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid, which is crucial for establishing rigidity of algebraic lattice actions on various geometric structures.

Contribution

It introduces a new non-stationary normal form theory and demonstrates local smooth rigidity for broad classes of algebraic actions of higher rank lattices.

Findings

Most homogeneous Anosov actions are locally smoothly rigid.

The new non-stationary normal form theory is a key technical advancement.

Rigidity results apply to actions on tori, nil-manifolds, and Furstenberg boundaries.

Abstract

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.