Global rigidity results for lattice actions on tori and new examples of volume-preserving actions

Any action of a finite index subgroup in SL(n, ℤ), n ≥ 4 on the n-dimensional torus which has a finite orbit and contains an Anosov element which splits as a direct product is smoothly conjugate to an affine action. We also construct first examples of real-analytic volume-preserving actions of SL(n, ℤ) and other higher-rank lattices on compact manifolds which are not conjugate (even topologically) to algebraic models.