Global smooth and topological rigidity of hyperbolic lattice actions

Abstract

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

Suppose $\Gamma$ is a lattice in 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 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.

Authors

Aaron Brown

University of Chicago, Chicago, IL 60637

Federico Rodriguez Hertz

Penn State University, University Park, State College, PA 16802

Zhiren Wang

Penn State University, University Park, State College, PA 16802