Global rigidity of actions by higher rank lattices with dominated splitting

TL;DR

This paper proves that smooth volume-preserving actions of certain higher rank lattices with dominated splitting on closed manifolds are standard, showing they are conjugate to affine actions on flat tori, establishing a form of global rigidity.

Contribution

It establishes a topological global rigidity result for actions of higher rank lattices with dominated splitting, extending known rigidity theorems to broader classes of manifolds and actions.

Findings

Actions are smoothly conjugate to affine actions on flat tori.

Global rigidity holds for actions with dominated splitting.

Similar results apply to lattices in symplectic and orthogonal groups.

Abstract

We prove that any smooth volume-preserving action of a lattice $\Gamma$ in $\textrm{SL}(n,\mathbb{R})$, $n\ge 3$, on a closed $n$-manifold which possesses one element that admits a dominated splitting should be standard. In other words, the manifold is the $n$-flat torus and the action is smoothly conjugate to an affine action. Note that an Anosov diffeomorphism, or more generally, a partial hyperbolic diffeomorphism admits a dominated splitting. We have a topological global rigidity when $\alpha$ is $C^{1}$. Similar theorems hold for an action of a lattice in $\textrm{Sp}(2n,\mathbb{R})$ with $n\ge 2$ and $\textrm{SO}(n,n)$ with $n\ge 5$ on a closed $2n$-manifold.