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The Zimmer Program for partially hyperbolic actions

Danijela Damjanovic, Ralf Spatzier, Kurt Vinhage, Disheng Xu

TL;DR

This work advances the Zimmer program by proving that volume-preserving actions of higher-rank semisimple groups and their lattices, subject to total partial hyperbolicity and accessibility, are smoothly conjugate (up to finite covers) to bi-homogeneous models on bi-quotients. It develops a two-pronged approach: (i) classify and rigidify totally partially hyperbolic $\mathbb{R}^k$-actions through invariant structures and the invariance principle, yielding local-to-global bi-homogeneous modeling; (ii) lift abelian dynamics to principal bundles to create leafwise homogeneous structures and then apply a topological/HAPHAs framework to obtain homogeneous models for the full $G$-action. The results unify Anosov and partially hyperbolic cases, extend rigidity beyond Cartan actions, and provide a comprehensive pathway from dynamical hypotheses to explicit homogeneous/bi-homogeneous classifications. Methodologically, the paper leverages Zimmer cocycle rigidity, the invariance principle, leafwise conformality, and novel bundle- and group-extension constructions, culminating in a global homogeneous description on compact manifolds and offering a platform for future Zimmer-program investigations. The findings have significant implications for understanding which manifolds support higher-rank group actions and how dynamical properties force algebraic structure in the Zimmer program context.

Abstract

Zimmer's superrigidity theorems on higher rank Lie groups and their lattices launched a program of study aiming to classify actions of semisimple Lie groups and their lattices, known as the {\it Zimmer program}. When the group is too large relative to the dimension of the phase space, the Zimmer conjecture predicts that the actions are all virtually trivial. At the other extreme, when the actions exhibit enough regular behavior, the actions should all be of algebraic origin. We make progress in the program by showing smooth conjugacy to a bi-homogeneous model (up to a finite cover) for volume-preserving actions of semisimple Lie groups without compact or rank one factors, which have two key assumptions: partial hyperbolicity for a large class of elements ({\it totally partial hyperbolicity}) and accessibility, a condition on the webs generated by dynamically-defined foliations. We also obtain classification for actions of higher-rank abelian groups satisfying stronger assumptions.

The Zimmer Program for partially hyperbolic actions

TL;DR

This work advances the Zimmer program by proving that volume-preserving actions of higher-rank semisimple groups and their lattices, subject to total partial hyperbolicity and accessibility, are smoothly conjugate (up to finite covers) to bi-homogeneous models on bi-quotients. It develops a two-pronged approach: (i) classify and rigidify totally partially hyperbolic -actions through invariant structures and the invariance principle, yielding local-to-global bi-homogeneous modeling; (ii) lift abelian dynamics to principal bundles to create leafwise homogeneous structures and then apply a topological/HAPHAs framework to obtain homogeneous models for the full -action. The results unify Anosov and partially hyperbolic cases, extend rigidity beyond Cartan actions, and provide a comprehensive pathway from dynamical hypotheses to explicit homogeneous/bi-homogeneous classifications. Methodologically, the paper leverages Zimmer cocycle rigidity, the invariance principle, leafwise conformality, and novel bundle- and group-extension constructions, culminating in a global homogeneous description on compact manifolds and offering a platform for future Zimmer-program investigations. The findings have significant implications for understanding which manifolds support higher-rank group actions and how dynamical properties force algebraic structure in the Zimmer program context.

Abstract

Zimmer's superrigidity theorems on higher rank Lie groups and their lattices launched a program of study aiming to classify actions of semisimple Lie groups and their lattices, known as the {\it Zimmer program}. When the group is too large relative to the dimension of the phase space, the Zimmer conjecture predicts that the actions are all virtually trivial. At the other extreme, when the actions exhibit enough regular behavior, the actions should all be of algebraic origin. We make progress in the program by showing smooth conjugacy to a bi-homogeneous model (up to a finite cover) for volume-preserving actions of semisimple Lie groups without compact or rank one factors, which have two key assumptions: partial hyperbolicity for a large class of elements ({\it totally partial hyperbolicity}) and accessibility, a condition on the webs generated by dynamically-defined foliations. We also obtain classification for actions of higher-rank abelian groups satisfying stronger assumptions.
Paper Structure (89 sections, 134 theorems, 157 equations, 3 figures)

This paper contains 89 sections, 134 theorems, 157 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Totally partially hyperbolic actions
  4. Bi-homogeneous actions
  5. Rigidity for semisimple group actions
  6. Rigidity for $\mathbb R^k$-actions
  7. Rigidity for Anosov $\mathbb R^k$ actions
  8. Rigidity for partially hyperbolic $\mathbb R^k$-actions.
  9. Questions and conjectures
  10. Rank one factors for $\mathbb{R}^k$-actions, Oseledets conformality, and a partially hyperbolic Katok-Spatzier conjecture
  11. Actions with one regular element
  12. Essential accessibility
  13. Systems with dominated splittings
  14. Products of groups of real rank 1
  15. Invariant Volumes
  16. ...and 74 more sections

Key Result

Theorem 2.5

Suppose that every simple factor of a real semisimple group $G$ has real rank at least 2, and let $G \curvearrowright X$ be a $C^\infty$ totally partially hyperbolic accessible action. Assume further that the restriction of the action to a maximal $\mathbb R-$split Cartan subgroup $A$ preserves an i

Figures (3)

  • Figure 1: Lyapunov exponents in $\Sigma(\alpha,\beta)$
  • Figure 2: Decomposing $(\mathbb{R}^k)^*$ into quadrants
  • Figure 3: A geometric commutator

Theorems & Definitions (303)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • Corollary 2.10
  • ...and 293 more