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Solvable groups acting on the line. (English) Zbl 0569.57012
Let G a group of homeomorphisms of $${\mathbb{R}}$$. The author shows that if G has a quasi-invariant measure in a strong sense and is minimal then it is conjugate to a subgroup of the affine group. Various conditions for the existence of a q.i. measure are given; this happens for G polycyclic or G analytic. For $$C^ 2$$ abelian groups the existence of an invariant measure is shown. Applications are given to codimension one foliations covered by an $${\mathbb{R}}$$-foliation and to Reeb stability questions. Also the author points out an improvement of an earlier theorem, implying Verjovsky’s conjecture: A codimension one Anosov flow on a manifold with solvable $$\Pi_ 1$$ is the suspension of a hyperbolic toral automorphism.
Reviewer: V.Sergiescu

##### MSC:
 57R30 Foliations in differential topology; geometric theory 37D99 Dynamical systems with hyperbolic behavior 57S20 Noncompact Lie groups of transformations 57S25 Groups acting on specific manifolds
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