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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 $$\mathbb{C}$$
##### Keywords:
Lie groups; dynamical systems; foliations; Anosov flows
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##### References:
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