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Actions localement libres du groupe affine. (French) Zbl 0577.57010
In this interesting paper, locally free actions of the group GA of orientation preserving affine transformations of $${\mathbb{R}}$$ on closed three-manifolds are considered. If M is a closed 3-dimensional manifold with $$H^ 1(M, {\mathbb{R}})=0$$, then any locally free $$C^ 2$$-action of GA on M preserves a $$C^ 0$$-volume form (Theorem D). Any locally free $$C^ r$$-action (r$$\geq 2)$$ of GA on any closed 3-manifold which preserves a $$C^ 0$$-volume form is $$C^{r-1}$$-conjugate to a ”homogeneous action”, i.e., an action of the form GA$$\times G/\Gamma \to G/\Gamma$$, where G is a Lie group containing GA as a subgroup, $$\Gamma$$ is a discrete uniform subgroup of G and the action is induced by left translations on G (Theorem B).
These are the main results of the article. However, it contains several other interesting results. Among the others: (1) homogeneous actions of GA on three manifolds are classified; (2) it is proved that any $$C^ 0$$- volume form preserved by a locally free $$C^ r$$-action of a non- unimodular Lie group G on a closed manifold M is of the class $$C^{r-2}$$ if only dim M$$=\dim G+1$$; (3) structural stability of some actions of fundamental groups of closed oriented surfaces of genus $$g\geq 2$$ on $$S^ 1$$ is established.
In the proofs, several deep results of the geometric theory of dynamical systems, ergodic theory and the theory of foliations are exploited.
Reviewer: P.Walczak

##### MSC:
 57R30 Foliations in differential topology; geometric theory 57S20 Noncompact Lie groups of transformations 37A99 Ergodic theory
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