Global smooth and topological rigidity of hyperbolic lattice actions

Aaron W Brown, Federico Rodriguez Hertz, Zhiren Wang

Research output: Contribution to journalArticle

1 Scopus citations

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

Keywords

  • Actions of higher-rank lattices
  • Anosov actions
  • Global rigidity
  • Smooth rigidity
  • Topological rigidity

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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