Global smooth and topological rigidity of hyperbolic lattice actions

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose Γ is a lattice in a 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^∞ action and contains an Anosov element, then the semiconjugacy is a C^∞ conjugacy. As a corollary, we obtain C^∞ global rigidity for Anosov actions by co- compact lattices in semisimple Lie groups with all factors rank 2 or higher. We also obtain global rigidity of Anosov actions of SL(n; Z) on Tⁿ for n ≥ 5 and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.