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November 2017 Global smooth and topological rigidity of hyperbolic lattice actions
Aaron Brown, Federico Rodriguez Hertz, Zhiren Wang
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Ann. of Math. (2) 186(3): 913-972 (November 2017). DOI: 10.4007/annals.2017.186.3.3


In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices.

Suppose $\Gamma$ is a lattice in a semisimple Lie group, all of whose factors have rank $2$ or higher. Let $\alpha$ be a smooth $\Gamma$-action on a compact nilmanifold $M$ that lifts to an action on the universal cover. If the linear data $\rho$ of $\alpha$ contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of $\alpha$ and $\rho$ on a finite-index subgroup of $\Gamma$. If $\alpha$ 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 groups 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 $n\ge 5$ and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.


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Aaron Brown. Federico Rodriguez Hertz. Zhiren Wang. "Global smooth and topological rigidity of hyperbolic lattice actions." Ann. of Math. (2) 186 (3) 913 - 972, November 2017.


Published: November 2017
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2017.186.3.3

Primary: 37C85 , 37D20
Secondary: 37C20 , 57S25

Keywords: actions of higher-rank lattices , Anosov actions , global rigidity , smooth rigidity , topological rigidity

Rights: Copyright © 2017 Department of Mathematics, Princeton University


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Vol.186 • No. 3 • November 2017
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