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Totally nonsymplectic Anosov actions on tori and nilmanifolds. (English) Zbl 1213.37043

To state the results we recall some notions and introduce some notation.
To a higher rank abelian group action on a manifold \(M\) is associated a splitting of the tangent bundle \(TM=\bigoplus_i E_i\) which refines the Lyapunov splitting of individual elements. For \(v\in E_i\), the Lyapunov exponent of \(v\) defines a linear functional called the Lyapunov functional, which the authors consider as a linear functional on the ambient \(\mathbb R^k\). A Weyl chamber is then a connected component of \(\mathbb R^k\) minus all the hyperplane kernels of the Lyapunov functionals.
A nilmanifold is a quotient \(N/\Gamma\) of a simply connected nilpotent Lie group \(N\) by a discrete cocompact subgroup \(\Gamma\). An action by affine automorphisms of a nilmanifold is semisimple if the linear part of every element acts by a semisimple matrix (that is, diagonalizable over \(\mathbb C\)).
A \(\mathbb Z^k\)-action is said to be totally nonsymplectic if any two elements \(v\in E_i\) and \(w\in E_j\) belong to the stable distribution of some element \(a\in \mathbb Z^k\).
The main result of the paper under review is the following global rigidity result for totally nonsymplectic actions.
Theorem. Let \(\alpha\) be a \(C^\infty\)-action of \(\mathbb Z^k\), \(k\geq 2\) on a nilmanifold \(N/\Gamma\). Assume that the linearization \(\rho\) of \(\alpha\) is semisimple and totally nonsymplectic, and there exists an Anosov element in each Weyl chamber of \(\alpha\). Then \(\alpha\) is \(C^{\infty}\)-conjugate to \(\rho\).
As a corollary, they obtain the following:
Corollary. Let \(\alpha\) be a \(C^\infty\)-action of \(\mathbb Z^k\), \(k\geq 2\) on a nilmanifold \(N/\Gamma\). Assume that the linearization \(\rho\) of \(\alpha\) is totally reducible and totally nonsymplectic and that there is an Anosov element in each Weyl chamber of \(\alpha\). Then \(\alpha\) is \(C^\infty\)-conjugate to \(\rho\).
The results obtained have applications to global rigidity for actions of higher rank lattices.

MSC:

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
53C24 Rigidity results
22E40 Discrete subgroups of Lie groups
22E46 Semisimple Lie groups and their representations
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