[Paper Review] Global smooth and topological rigidity of hyperbolic lattice actions

Aaron Brown, Federico Rodriguez Hertz+1 more|arXiv (Cornell University)|Dec 21, 2015

Mathematical Dynamics and Fractals27 references3 citations

TL;DR

This paper establishes global smooth and topological rigidity for hyperbolic lattice actions on compact nilmanifolds. It proves that under hyperbolic linear data or Anosov dynamics, there exists a continuous semiconjugacy (and $C^\infty$ conjugacy if smooth and Anosov), leading to $C^\infty$ global rigidity for cocompact lattices in higher-rank semisimple Lie groups and for $\mathrm{SL}(n,\mathbb{Z})$ on $\mathbb{T}^n$ ($n \geq 5$).

ABSTRACT

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose $Γ$ is a lattice in 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 $C^\infty$ action and contains an Anosov element, then the semiconjugacy is a $C^\infty$ conjugacy. As a corollary, we obtain $C^\infty$ global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank $2$ or higher. We also obtain global rigidity of Anosov actions of $\mathrm{SL}(n,\mathbb Z)$ on $\mathbb T^n$ for $ n\geq 5$ and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.

Motivation & Objective

• To establish global topological and smooth rigidity for higher-rank lattice actions on compact nilmanifolds with hyperbolic or Anosov dynamics.
• To prove that such actions are semiconjugate (or conjugate in the smooth case) to their linear data, extending Zimmer program conjectures.
• To resolve the lifting problem of actions on nilmanifolds via cohomological vanishing of defect functionals.
• To establish $C^\infty$ global rigidity for Anosov actions of cocompact lattices in higher-rank semisimple Lie groups.
• To prove rigidity for $\mathrm{SL}(n,\mathbb{Z})$ actions on $\mathbb{T}^n$ for $n \geq 5$ and for probability-preserving Anosov actions on nilmanifolds.

Proposed method

• Construct a continuous semiconjugacy between the action $\alpha$ and its linear data $\rho$ using approximate conjugacy and control of central defects.
• Use the lifting property and $\pi_1$-factors to reduce the problem to nilpotent group actions with controlled defect functionals.
• Define the defect functional $\beta_i(\gamma_1, \gamma_2)$ measuring failure of lifts to commute, with values in $Z_i \cap \Lambda_i$.
• Prove that the defect functional vanishes in cohomology $H^2_{\rho_i}(\Gamma; \mathbb{Z}^{d_i})$ by showing $d_{\rho_i,1}\eta = \beta_i$ via integration against invariant measure.
• Pass to a finite-index subgroup $\hat{\Gamma}$ to lift the action coherently, enabling construction of the semiconjugacy.
• Apply cocycle superrigidity and orbit closure results to control dynamics and ensure regularity of the conjugacy.

Experimental results

Research questions

• RQ1 Under what conditions does a smooth lattice action on a nilmanifold admit a $C^\infty$ conjugacy to its linear data?
• RQ2 When does a hyperbolic lattice action on a nilmanifold admit a continuous semiconjugacy to its linear representation?
• RQ3 Can the lifting obstruction for nilmanifold actions be resolved via vanishing of cohomological defect functionals?
• RQ4 What is the global rigidity class of Anosov actions of $\mathrm{SL}(n,\mathbb{Z})$ on $\mathbb{T}^n$ for $n \geq 5$?
• RQ5 Do probability-preserving Anosov actions of higher-rank lattices on nilmanifolds admit $C^\infty$ global rigidity?

Key findings

• For a smooth $\Gamma$-action on a compact nilmanifold with hyperbolic linear data, there exists a continuous semiconjugacy to the linear action on a finite-index subgroup of $\Gamma$.
• If the action is $C^\infty$ and contains an Anosov element, the semiconjugacy becomes a $C^\infty$ conjugacy.
• The defect functional $\beta_i$ vanishes in $H^2_{\rho_i}(\Gamma; \mathbb{Z}^{d_i})$ after passing to a finite-index subgroup, enabling coherent lifting of the action.
• The $C^\infty$ global rigidity holds for Anosov actions of cocompact lattices in semisimple Lie groups with all factors of rank $\geq 2$.
• The paper establishes $C^\infty$ global rigidity for $\mathrm{SL}(n,\mathbb{Z})$ actions on $\mathbb{T}^n$ for $n \geq 5$.
• Probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds are $C^\infty$ globally rigid.