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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

22D40 Ergodic theory on groups
28D15 General groups of measure-preserving transformations
Full Text: DOI
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