Global rigidity of actions by higher rank lattices with dominated splitting

Abstract

We prove that any smooth volume-preserving action of a lattice $\Gamma$ in $\text{SL}(n,\mathbb{R})$, $n\ge 3$, 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 $\text{Sp}(2n,\mathbb{R})$ with $n\ge 2$ and $\text{SO}(n,n)$ with $n\ge 5$ on a closed $2n$-manifold.

arXiv commentMinor changes. (The title is changed.)

References in corpus (1)

• Groups acting on manifolds: around the Zimmer program