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Locally free actions of solvable Lie groups. (Actions localement libres de groupes résolubles.) (French) Zbl 0816.58021
Summary: Let \(G\) be a connected \((n-1)\)-dimensional Lie group and \(\Phi\) a \(C^ r (r \geq 2)\) locally free action of \(G\) on a compact \(n\)-dimensional manifold (\(n \geq 3\)). We assume that the Lie algebra of \(G\) contains a field \(Y\) such that the eigenvalues of \(\text{ad}(Y)\) are \(\alpha_ 1,...,\alpha_{n-2},0\) with \(\text{Re}(\alpha_ i)<0\). Then, we show that \(\Phi\) is \(C^ r\)-conjugated to a homogeneous action of \(G\) on \(H/\Gamma\) where \(H\) is a Lie group containing \(H\) and \(\Gamma\) a lattice of \(H\). We provide many examples related to Anosov theory.

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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