Global smooth and topological rigidity of hyperbolic lattice actions

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose Γ is a lattice in a semisimple Lie group, all of whose factors have rank 2 or higher. Let α be a smooth Γ-action on a compact nilmanifold M that lifts to an action on the universal cover. If the linear data ρ of α contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of α and ρ on a finite-index subgroup of Γ. If α is a C action and contains an Anosov element, then the semiconjugacy is a C conjugacy. As a corollary, we obtain C global rigidity for Anosov actions by co- compact lattices in semisimple Lie groups with all factors rank 2 or higher. We also obtain global rigidity of Anosov actions of SL(n; Z) on Tn for n ≥ 5 and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.

Original languageEnglish (US)
Pages (from-to)913-972
Number of pages60
JournalAnnals of Mathematics
Volume186
Issue number3
DOIs
StatePublished - Nov 1 2017

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Rigidity
Nilmanifolds
Semisimple Lie Group
Universal Cover
Conjugacy
Corollary
Subgroup
Arbitrary

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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abstract = "In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose Γ is a lattice in a semisimple Lie group, all of whose factors have rank 2 or higher. Let α be a smooth Γ-action on a compact nilmanifold M that lifts to an action on the universal cover. If the linear data ρ of α contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of α and ρ on a finite-index subgroup of Γ. If α is a C∞ action and contains an Anosov element, then the semiconjugacy is a C∞ conjugacy. As a corollary, we obtain C∞ global rigidity for Anosov actions by co- compact lattices in semisimple Lie groups with all factors rank 2 or higher. We also obtain global rigidity of Anosov actions of SL(n; Z) on Tn for n ≥ 5 and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.",
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Global smooth and topological rigidity of hyperbolic lattice actions. / Brown, Aaron; Rodriguez Hertz, Federico Juan; Wang, Zhiren.

In: Annals of Mathematics, Vol. 186, No. 3, 01.11.2017, p. 913-972.

Research output: Contribution to journalArticle

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