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Global Rigidity of Codimension One Actions

Camilo Arosemena Serrato

TL;DR

The paper proves a global rigidity result for smooth locally free codimension-one actions of a higher-rank simple split group $G$ under a mixing $P$-invariant measure. The strategy blends higher-rank ergodic theory, Pesin theory with transverse directions, and a topological-measure rigidity framework to produce a saturated transverse submanifold $N$ and a smooth equivariant map to $G/Q$, yielding either a suspension structure or a finite cover by $G/\Gamma\times S^1$. Key innovations include integrating stable and center-unstable foliations in the transverse direction, and leveraging harmonic/stationary measures to constrain ergodic components and deduce algebraic models. The results extend Nevo–Zimmer’s measurable rigidity to a smooth setting, connecting the Zimmer program with Deroin–Hurtado and related entropy methods to identify algebraic origins for broad classes of actions.

Abstract

Consider a smooth, locally free, codimension-one action of a higher-rank, simple, split Lie group $G$ on a closed manifold $M$. Let $P$ be a minimal parabolic subgroup of $G$. If the action admits a $P$-invariant probability measure that is mixing, then the action is either equivariantly diffeomorphic to the suspension of a codimension one, locally free action on a closed manifold of a parabolic subgroup of $G$; or, it is finitely and equivariantly covered by the action of $G$ on $G/Γ\times S^1$, where the action on $G/Γ$ is the coset action, and $G$ acts trivially on $S^1$. We prove this by doing a jointly integration argument of stable and center unstable Pesin manifolds. This is a smooth version of results by Nevo and Zimmer.

Global Rigidity of Codimension One Actions

TL;DR

The paper proves a global rigidity result for smooth locally free codimension-one actions of a higher-rank simple split group under a mixing -invariant measure. The strategy blends higher-rank ergodic theory, Pesin theory with transverse directions, and a topological-measure rigidity framework to produce a saturated transverse submanifold and a smooth equivariant map to , yielding either a suspension structure or a finite cover by . Key innovations include integrating stable and center-unstable foliations in the transverse direction, and leveraging harmonic/stationary measures to constrain ergodic components and deduce algebraic models. The results extend Nevo–Zimmer’s measurable rigidity to a smooth setting, connecting the Zimmer program with Deroin–Hurtado and related entropy methods to identify algebraic origins for broad classes of actions.

Abstract

Consider a smooth, locally free, codimension-one action of a higher-rank, simple, split Lie group on a closed manifold . Let be a minimal parabolic subgroup of . If the action admits a -invariant probability measure that is mixing, then the action is either equivariantly diffeomorphic to the suspension of a codimension one, locally free action on a closed manifold of a parabolic subgroup of ; or, it is finitely and equivariantly covered by the action of on , where the action on is the coset action, and acts trivially on . We prove this by doing a jointly integration argument of stable and center unstable Pesin manifolds. This is a smooth version of results by Nevo and Zimmer.

Paper Structure

This paper contains 59 sections, 57 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Acknowledgments
  4. Preliminaries
  5. The theorem of Pugh and Shub on $\mathbb R^k$ ergodic measures
  6. Lyapunov exponents and Pesin manifolds
  7. Osedelec's multiplicative ergodic theorem and Lyapunov exponents
  8. Unstable Pesin manifolds
  9. Entropy and The Ledrappier-Young theorem
  10. Entropy conditional on measurable foliations
  11. Conditional entropy along measurable partitions and the Ledrappier-Young Theorem
  12. Geometric characterization of the defect in the entropy formula
  13. Smooth ergodic theory of higher rank abelian actions
  14. Higher-rank Osedelec's Theorem
  15. Global unstable Pesin manifolds in higher rank actions
  16. ...and 44 more sections

Key Result

Theorem 1.1

Suppose $G$ acts locally freely and in a $C^2$ manner on a closed connected manifold $M$, with codimension $1$ orbits. Suppose there exists a $P$-invariant probability measure $\mu^P$ on $M$ such that $\ker(\alpha)\curvearrowright (M,\mu^P)$ is ergodic, for all $\alpha\in\Pi_P$. Then exactly one of

Theorems & Definitions (108)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2: Osedelec osedelec
  • Theorem 2.3: Ruelle-Margulis inequality
  • Theorem 2.4: Ledrappier-Young theorem, ledrappieryoung1
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7: Higher Rank Osedelec's Theorem
  • Definition 2.8
  • Lemma 2.9
  • ...and 98 more