  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.
