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Rigidity for Anosov actions of higher rank lattices. (English) Zbl 0754.58029
Let $$G$$ be a connected semisimple algebraic $$\mathbb{R}$$-group with finite center and $$G^ 0_ \mathbb{R}$$ having no compact factors. A finitely generated discrete subgroup $$\Gamma\subset G$$ such that $$G/\Gamma$$ has a finite volume and the $$\mathbb{R}$$-split rank of each factor of $$G$$ is at least 2 is called a higher rank lattice.
The author studies the rigidity and deformation rigidity of $$C^ k$$- actions $$\varphi: \Gamma\times X\to X$$ on a compact manifold $$X$$ under the assumption that an appropriate element $$\gamma\in\Gamma$$ is an Anosov diffeomorphism (i.e., a continuous splitting $$TX=E^ +\oplus E^ -$$ exists such that $$\| D(\gamma^ m)\|>c\lambda^ m$$ on $$E^ +$$, $$\| D(\gamma^ m)\|<1/c \lambda^ m$$ on $$E^ -$$ for certain $$c>0$$, $$\lambda>1$$). By analysing the behavior of periodic orbits and with additional hypothesis on the first cohomology of $$\Gamma$$ ensuring their stability under perturbations, the topological conjugacy ($$r=0$$) of full group action is proved. Then the criteria when the topological conjugacy implies the smooth conjugacy are derived by employing the concept of a trellised Cartan action (in rough, the existence of a certain family of one-dimensional foliations invariant under a volume- preserving free abelian subgroup $${\mathcal A}\subset\Gamma$$ is postulated with $$\text{rank }{\mathcal A}=\dim X$$). Numerous applications of definite nature on actions of various subgroups $$\Gamma\subset \text{SL}(n,\mathbb{Z})$$ on the torus $$\mathbb{T}^ n=\mathbb{R}^ n/\mathbb{Z}^ n$$ are presented.
Reviewer: J.Chrastina (Brno)

##### MSC:
 37D99 Dynamical systems with hyperbolic behavior 58H15 Deformations of general structures on manifolds 22E40 Discrete subgroups of Lie groups
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