Extensions of generic measure-preserving actions

Melleray, Julien

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 607-623.

Nous établissons que, pour tout groupe dénombrable abélien $\Gamma$ et tout sous-groupe finiment engendré $\Delta$ de $\Gamma$, l’ensemble des actions de $\Delta$ sur un espace de probabilités standard $(X,\mu )$ qui peuvent être étendues en une action libre de $\Gamma$ sur $(X,\mu )$ est générique (au sens de Baire). Ce résultat étend un théorème d’Ageev, qui correspond au cas où $\Delta$ est un groupe cyclique infini.

We show that, whenever $\Gamma$ is a countable abelian group and $\Delta$ is a finitely-generated subgroup of $\Gamma$, a generic measure-preserving action of $\Delta$ on a standard atomless probability space $(X,\mu )$ extends to a free measure-preserving action of $\Gamma$ on $(X,\mu )$. This extends a result of Ageev, corresponding to the case when $\Delta$ is infinite cyclic.

Classification : 22F10, 54H05, 54E52
Keywords: Measure-preserving action, Baire category, Polish group
Mot clés : action préservant une mesure de probabilité, méthodes de Baire, groupe polonais

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