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Locally compact groups appearing as ranges of cocycles of ergodic $${\mathbb{Z}}$$-actions. (English) Zbl 0574.22005
Let G,$$\Gamma$$ be separable, locally compact groups, let $$\Gamma$$ act ergodically on a Lebesgue space (X,$$\mu)$$, and let $$\alpha$$ : $$X\times \Gamma \to G$$ be a 1-cocycle of this action. Then one can construct an action of $$\Gamma$$ on (X$$\times G, \mu \times m)$$ (m $$=$$ right Haar measure on G) by $$(x,g)\cdot \gamma =(x\cdot \gamma, g\alpha (x,\gamma))$$. If this action is ergodic, G is said to be the range of $$\alpha$$. R. Zimmer has shown that if G is the range of a cocycle of an ergodic $${\mathbb{Z}}$$-action, then every ergodic action of G is a Poincaré flow (i.e., is the range of a cocycle of an ergodic automorphism). The authors, improving on results of Zimmer, show that every amenable, almost connected G is the range of a cocycle of an ergodic $${\mathbb{Z}}$$-action.
Reviewer: L.Corwin

##### MSC:
 22D40 Ergodic theory on groups 28D15 General groups of measure-preserving transformations
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##### References:
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