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 Gamma is a lattice in a semisimple Lie group, all of whose factors have rank 2 or higher. Let a be alpha 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-infinity action and contains an Anosov element, then the semiconjugacy is a C-infinity conjugacy. As a corollary, we obtain C-infinity 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 T-n for n >= 5 and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds.