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Global rigidity of higher rank lattice actions with dominated splitting
Published online by Cambridge University Press: 27 April 2023
Abstract
Let $\alpha $ be a
$C^{\infty }$ volume-preserving action on a closed n-manifold M by a lattice
$\Gamma $ in
$\mathrm {SL}(n,\mathbb {R})$,
$n\ge 3$. Assume that there is an element
$\gamma \in \Gamma $ such that
$\alpha (\gamma )$ admits a dominated splitting. We prove that the manifold M is diffeomorphic to the torus
${{\mathbb T}^{n}={\mathbb R}^{n}/{\mathbb Z}^{n}}$ and
$\alpha $ is smoothly conjugate to an affine action. Anosov diffeomorphisms and partial hyperbolic diffeomorphisms admit a dominated splitting. We obtained a topological global rigidity when
$\alpha $ is
$C^{1}$. We also prove similar theorems for actions on
$2n$-manifolds by lattices in
$\textrm {Sp}(2n,{\mathbb R})$ with
$n\ge 2$ and
$\mathrm {SO}(n,n)$ with
$n\ge 5$.
Keywords
MSC classification
- Type
- Original Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press
References
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