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Global smooth and topological rigidity of hyperbolic lattice actions. (English) Zbl 1379.37060
The authors study the global rigidity problem for actions of higher-rank lattices on nilmanifolds with hyperbolic linear data. They provide complete solutions to some global rigidity questions under the mild assumption that the action lifts to an action on the universal covering. For such actions with hyperbolic linear data, the authors construct a continuous semiconjugacy to the linear data when restrited to a finite-index subgroup. In particular, if the action contains an Anosov element, they show that the semiconjugacy is in fact a $$C^\infty$$-diffeomorphism.

##### MSC:
 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37C20 Generic properties, structural stability of dynamical systems 57S25 Groups acting on specific manifolds
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