Rigidity for Anosov actions of higher rank lattices.

*(English)*Zbl 0754.58029Let \(G\) 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.

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)