An uncountable family of nonorbit equivalent actions of 𝔽_{𝕟}

Gaboriau, Damien; Popa, Sorin

doi:10.1090/S0894-0347-05-00480-7

An uncountable family of nonorbit equivalent actions of $\mathbb {F}_n$
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by Damien Gaboriau and Sorin Popa

J. Amer. Math. Soc. 18 (2005), 547-559

DOI: https://doi.org/10.1090/S0894-0347-05-00480-7

Published electronically: March 28, 2005

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Abstract:

For each $2 \leq n \leq \infty$, we construct an uncountable family of free ergodic measure preserving actions $\alpha _t$ of the free group $\mathbb {F}_n$ on the standard probability space $(X, \mu )$ such that any two are nonorbit equivalent (in fact, not even stably orbit equivalent). These actions are all “rigid” (in the sense of Popa), with the II$_1$ factors $L^\infty (X, \mu )\rtimes _{\alpha _t} \mathbb {F}_n$ mutually nonisomorphic (even nonstably isomorphic) and in the class $\mathcal {H}\mathcal {T}_{_{s}}.$

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